1. The area of a rectangle can illustrate the distributive property. If the squares original length is "x" the equation would be x^2. Then if you wanted to add 7 to one side and 4 to the other the equation would be A=(x+7)(x+4), A being the area. Then you distribute to get the area:
A=(x+7)x+(x+7)4
A=x^2+7x+4x+28
A=x^2+11x+28
2.a. If a quadratic expression is in factored form and you want it in expanded form you do the following:
A=(x+9)(x+6)
A=(x+9)x+(x+9)6 Use the distributive property to multiply (x+9) by x and 6.
A=x^2+9x+6x+54 Distribute.
A=x^2+15x+54 Combine like terms. (you don't have to)
2.b. If a quadratic expression is in expanded form and you want it it factored form do the following:
A=x^2+7x+10
A=x^2+5x+2x+10 Find two numbers that add up to seven and multiply to 10.
A=(x+5)(x+2) Put into factored form.
3. You can recognize a quadratic function from it's equation because in a quadratic function's equation there is always and x^2 (for expanded form) or two X's being multiplied (for factored form).
4. Features of a quadratic function's graph are:
*Parabola(u shaped)
*Symmetrical-to find the equation for the line of symmetry is the number exactly between the two x-intercepts. The equation would be x=a.
*Has a maximum/minimum point-the x coordinate would be the number exactly in between the two x-intercepts and then you substitute the x in the equation to find the y coordinate.
*x-intercept(s)-these are the oposites of the two constants in the factored equation. If there is only one constant the other is 0.
*y-intercept-the constant or the number with no variable in the expanded equation.
Sunday, May 16, 2010
Monday, May 10, 2010
Math Reflection 2
1. The area of a rectangle can illustrate the distributive property in several ways. First, one side of the rectangle could be broken up into different segments and the other sides value ( such as x) could me multiplied by the addition of the different segments. Also, both sides could be divided into different segments and the addition of the segments of one side could be distributed on the addition of the segments of the other side.
2.a. If a quadratic expression is in factored form you can change it to factored form by distributing one of the segments on the addition of the other side's segments, and then distributing the other segment on the combination of the other side.
2.b. You change a quadratic expression which is in expanded form to an expression in factored form by first figuring out which two numbers divide the variable by so that the addition of both numbers adds up to what it was originally was and can multiply to get the number in parentheses. For instance 6x+9=3x+3x+9. The next step is to find the common number or variable that go into each segment and multiply it by the segments of that side divided by that specific number or variable and add it to the same thing for the other side. For instance 3x+6=3(x+2) The final step is to distribute it so that each set of parentheses have both a variable and a number and the the parentheses are multiplied by each other.
3. You know that an equation in factored form is quadratic if the highest exponent is exactly 2. You know that an equation in expanded form is quadratic if it has 2 variables both raised to the first power.
4. The shape of a graph of a quadratic expression is always a parabola. This means that it looks like a hill or and upside down hill. If it opens down the slope starts high and gets smaller as x gets higher until the slope becomes negative and changes direction the negative slope increasing. It is the other way around for a slope that opens up. The graph also usually has two x intercepts unless the x intercept is the graphs max point (for a graph that opens down) or the x intercept is the minimum point (for a graph that opens up). Finally, the graph of a quadratic equation is symmetrical mirroring it self once it hits its max or minimum point.
2.a. If a quadratic expression is in factored form you can change it to factored form by distributing one of the segments on the addition of the other side's segments, and then distributing the other segment on the combination of the other side.
2.b. You change a quadratic expression which is in expanded form to an expression in factored form by first figuring out which two numbers divide the variable by so that the addition of both numbers adds up to what it was originally was and can multiply to get the number in parentheses. For instance 6x+9=3x+3x+9. The next step is to find the common number or variable that go into each segment and multiply it by the segments of that side divided by that specific number or variable and add it to the same thing for the other side. For instance 3x+6=3(x+2) The final step is to distribute it so that each set of parentheses have both a variable and a number and the the parentheses are multiplied by each other.
3. You know that an equation in factored form is quadratic if the highest exponent is exactly 2. You know that an equation in expanded form is quadratic if it has 2 variables both raised to the first power.
4. The shape of a graph of a quadratic expression is always a parabola. This means that it looks like a hill or and upside down hill. If it opens down the slope starts high and gets smaller as x gets higher until the slope becomes negative and changes direction the negative slope increasing. It is the other way around for a slope that opens up. The graph also usually has two x intercepts unless the x intercept is the graphs max point (for a graph that opens down) or the x intercept is the minimum point (for a graph that opens up). Finally, the graph of a quadratic equation is symmetrical mirroring it self once it hits its max or minimum point.
Sunday, May 9, 2010
Math Reflection 2
1) The area of a rectangle can illustrate the Distributive Property. For example, if you had a square with lengths x and you add 3 to one side and subtract 2 from the other side your equation would be A=(x+3)(x-2). If you change the rectangle at all it can be shown in the equation. If you double one side the equation would be A=2x(x). No matter how you change it you will be able to recognize the difference in the equation with Distributive Property.
2a) If a quadratic expression is in factored form, to find an equivalent expression in expanded form all you have to do is do it out using distributive property. For instance, the equation in factored form is A=x(x+7)-5. You would do A=(x(x)+x(7))-5. After that the equation is A=x^2+7x-5, and that is the equation in expanded form.
2b) If a quadratic expression is in expanded form, to find and equivalent expression in factored form you use distributive property backwards. First you the common variable, then you put that out of the parentheses and put the other numbers or variables divided by the common factor in the parentheses and you have your equation. For example, the expanded equation is A=X^2-4x. The common variable is x. X^2/x=x, -4x/x=-4. You now have the numbers and variables that go inside and outside of the parentheses. The final equation is A=x(x-4).
3) You can recognize a quadratic function from its equation because in expanded form there will be a variable squared, or in factored form there will be a variable times itself.
4) The shape of a graph of a quadratic function is a parabola. It is u-shaped or upside down u-shaped. There are no straight lines and there are maximum and minimum points. There are two x-intercepts, and one y-intercept. There are also lines of symmetry whose equations usually equal x=x coordinate of maximum/minimum points.
2a) If a quadratic expression is in factored form, to find an equivalent expression in expanded form all you have to do is do it out using distributive property. For instance, the equation in factored form is A=x(x+7)-5. You would do A=(x(x)+x(7))-5. After that the equation is A=x^2+7x-5, and that is the equation in expanded form.
2b) If a quadratic expression is in expanded form, to find and equivalent expression in factored form you use distributive property backwards. First you the common variable, then you put that out of the parentheses and put the other numbers or variables divided by the common factor in the parentheses and you have your equation. For example, the expanded equation is A=X^2-4x. The common variable is x. X^2/x=x, -4x/x=-4. You now have the numbers and variables that go inside and outside of the parentheses. The final equation is A=x(x-4).
3) You can recognize a quadratic function from its equation because in expanded form there will be a variable squared, or in factored form there will be a variable times itself.
4) The shape of a graph of a quadratic function is a parabola. It is u-shaped or upside down u-shaped. There are no straight lines and there are maximum and minimum points. There are two x-intercepts, and one y-intercept. There are also lines of symmetry whose equations usually equal x=x coordinate of maximum/minimum points.
Math Reflection 2
1. You can show distributive property in a rectangle in many different ways. For example, if you add 2 to one side and 4 to another, the equation would be A=(x+2)(x+4). From the formula A=L(w), (x+2) is the length or L and (x+4) is the width or w. When you distribute and solve, you will get the expanded form of this same factored equation.
Step 1. Distribute the length of (x+2). It doesn't really matter which part you distribute.
A= x(x+4)+2(x+4) or x(x)+x(4)+2(x)+2(4)
Step 2. Simplify
A=x^2+4x+2x+8
Step 3. Add like terms.
A=x^2+6x+8
2a. To get the equivalent expression from factored form to expanded form, use the distributive property like the example above. Distribute, Simplify, and finally Add all like terms. Notice above as each step you do, the more it looks like the expanded form. of ax^2+bx+c.
2b. To get the equivalent expression from expanded form to factored form, you almost trace the opposite steps of the distibutive property. First, you have to know that the numbers added to the x, multipied is the "c" in the expanded form. Also, the sum of the numbers added to the x, is the "b" in the expanded form. For example, let's take x^2+8x+12. What two numbers has a sum of 8 AND has a product of 12?
Step 1. List all the factors of "c", in this case 12.
(1,12) (2,6) (3,4)
Step 2. Add up each pair and see what equals "b", or in this case 8.
1+12=13, 2+6=8, 3+4=7.
Step 3. When you find the two numbers, make the expanded form into four values. This will be the Step 2, when you turned the factored form into expanded form. This shows that you really are working backward.
x^2+2x+6x+12
Step 4. Now the two coefficents of x in the middle are your numbers that add up to x in the factored form.
(x+2)(x+6)
3. You can recognize a quadratic function from its equation very easily. From the expanded form, there is always a variable squared. In the factored form, there can and has to only have one variable per factor. Be careful, not all equations with exponents are quadratics. Exponential equations have the variable as the exponent, and quadratics has the variable for the base.
4. The shape of the graph is always a curved or parabola shape. The x-intercepts, y-intercepts, minimum or maximum point, and the line of symmetry is very important features to this graph.
You can find the x-intercepts in the factored form. Y will equal 0, which means one of the factors have to equal 0. So, take the opposite of each number added to the variables, and they will be your x-intercepts.
You can find the y-intercept by making x=0 and solving the equation. The easy or quick way to find the y-intercept is that you can look at the last value or "c" in the expanded form.
You can find the minimum or maximum point by taking the average of the x-intercepts and plugging it into the equation and solving.
You can find the line of symetry is by the x in the minimum or maximum point. Since it is a vertical line, the slope is undefined, so the equation will always be x= the average of the x-intercepts.
Step 1. Distribute the length of (x+2). It doesn't really matter which part you distribute.
A= x(x+4)+2(x+4) or x(x)+x(4)+2(x)+2(4)
Step 2. Simplify
A=x^2+4x+2x+8
Step 3. Add like terms.
A=x^2+6x+8
2a. To get the equivalent expression from factored form to expanded form, use the distributive property like the example above. Distribute, Simplify, and finally Add all like terms. Notice above as each step you do, the more it looks like the expanded form. of ax^2+bx+c.
2b. To get the equivalent expression from expanded form to factored form, you almost trace the opposite steps of the distibutive property. First, you have to know that the numbers added to the x, multipied is the "c" in the expanded form. Also, the sum of the numbers added to the x, is the "b" in the expanded form. For example, let's take x^2+8x+12. What two numbers has a sum of 8 AND has a product of 12?
Step 1. List all the factors of "c", in this case 12.
(1,12) (2,6) (3,4)
Step 2. Add up each pair and see what equals "b", or in this case 8.
1+12=13, 2+6=8, 3+4=7.
Step 3. When you find the two numbers, make the expanded form into four values. This will be the Step 2, when you turned the factored form into expanded form. This shows that you really are working backward.
x^2+2x+6x+12
Step 4. Now the two coefficents of x in the middle are your numbers that add up to x in the factored form.
(x+2)(x+6)
3. You can recognize a quadratic function from its equation very easily. From the expanded form, there is always a variable squared. In the factored form, there can and has to only have one variable per factor. Be careful, not all equations with exponents are quadratics. Exponential equations have the variable as the exponent, and quadratics has the variable for the base.
4. The shape of the graph is always a curved or parabola shape. The x-intercepts, y-intercepts, minimum or maximum point, and the line of symmetry is very important features to this graph.
You can find the x-intercepts in the factored form. Y will equal 0, which means one of the factors have to equal 0. So, take the opposite of each number added to the variables, and they will be your x-intercepts.
You can find the y-intercept by making x=0 and solving the equation. The easy or quick way to find the y-intercept is that you can look at the last value or "c" in the expanded form.
You can find the minimum or maximum point by taking the average of the x-intercepts and plugging it into the equation and solving.
You can find the line of symetry is by the x in the minimum or maximum point. Since it is a vertical line, the slope is undefined, so the equation will always be x= the average of the x-intercepts.
Math Reflection
1. The area of a rectangle can demonstrate the distributive property. If a square's length was originally x inches, and you added 3 inches to both the length and width, the equation for its area would be A = (x + 3) (x + 3), A being the area, because length x width = A. Currently, the equation is in factor form. If you want to convert it to expanded form, you use the distributive property like so:
A = (x + 3) (x + 3)
A = x (x) + x(3) + 3(x) + 3(3)
A = x^2 + 3x + 3x + 9
A = x^2 + 6x + 9
2.a. If a quadratic expression is in factored form, like y = (x + 3) (x + 2), you can find an equivalent expression in expanded form by using the distributive property:
y = (x + 3) (x + 2)
y = x(x) + x(2) + 3(x) + 3(2)
y = x^2 + 2x + 3x + 6
y = x^2 + 5x + 6
b. If a quadratic function is in expanded form, like y = x^2 + 5x + 6, you can find its equivalent expression in factor form by finding two numbers that, when you multiply them, equal the constant in the equation, and when you add them, equal the coefficient in the equation. The equivalent expression of y = x^2 + 5x + 6 is y = (x + 5) (x + 1), because then you need to substitute these numbers into the equation y = (x + n) (x + a), where n is one constant and a is the other. y = x^2 + 5x + 6 is equal to y = (x + 5) (x + 1), because 5 + 1 = 6 and 5 x 1 = 5.
3. An expression in factored form is quadratic if it has two linear factors with their variables raised to the first power. An expression in expanded form is quadratic if the variable's highest power is 2. x (x + 2) and x^2 + 2x are quadratic expressions.
4. The graph of a quadratic function always forms a parabola. It has a high or low point, 2 x intercepts, and a line of symmetry. Here is how you find each of these features from a quadratic function's equation:
x-intercepts - the opposites of the two constants in the factored form of the equation. If there is only one, then the other is 0. Before you find the opposites of the constants, you have to simplify the equation to y = (x + n) (x + a), where n and a are the constants.
y-intercept - the constant in the expanded form of the equation.
minimum/maximum point - to find the x-coordinate, find the number exactly between the x-intercepts, then substitute this number into the equation to find the y-coordinate.
equation of the line of symmetry - the number exactly between the x-intercepts is a in the equation x = a.
A = (x + 3) (x + 3)
A = x (x) + x(3) + 3(x) + 3(3)
A = x^2 + 3x + 3x + 9
A = x^2 + 6x + 9
2.a. If a quadratic expression is in factored form, like y = (x + 3) (x + 2), you can find an equivalent expression in expanded form by using the distributive property:
y = (x + 3) (x + 2)
y = x(x) + x(2) + 3(x) + 3(2)
y = x^2 + 2x + 3x + 6
y = x^2 + 5x + 6
b. If a quadratic function is in expanded form, like y = x^2 + 5x + 6, you can find its equivalent expression in factor form by finding two numbers that, when you multiply them, equal the constant in the equation, and when you add them, equal the coefficient in the equation. The equivalent expression of y = x^2 + 5x + 6 is y = (x + 5) (x + 1), because then you need to substitute these numbers into the equation y = (x + n) (x + a), where n is one constant and a is the other. y = x^2 + 5x + 6 is equal to y = (x + 5) (x + 1), because 5 + 1 = 6 and 5 x 1 = 5.
3. An expression in factored form is quadratic if it has two linear factors with their variables raised to the first power. An expression in expanded form is quadratic if the variable's highest power is 2. x (x + 2) and x^2 + 2x are quadratic expressions.
4. The graph of a quadratic function always forms a parabola. It has a high or low point, 2 x intercepts, and a line of symmetry. Here is how you find each of these features from a quadratic function's equation:
x-intercepts - the opposites of the two constants in the factored form of the equation. If there is only one, then the other is 0. Before you find the opposites of the constants, you have to simplify the equation to y = (x + n) (x + a), where n and a are the constants.
y-intercept - the constant in the expanded form of the equation.
minimum/maximum point - to find the x-coordinate, find the number exactly between the x-intercepts, then substitute this number into the equation to find the y-coordinate.
equation of the line of symmetry - the number exactly between the x-intercepts is a in the equation x = a.
Math Reflection 2
1. The side lengths of a rectangle might be x+4, and x+6. To find out the total area, you would multiply each value by eachother. You could use to equation (x+4)(x+6). It would come out to x^2 + 4x + 6x + 24, or x^2 + 10x + 24.
2a. You can use the Distributive Property to change a quadratic expression from its factored form into its expanded form. A quadratic expression in factored form can look like (x+3)(x+5). To change it to its expanded form, you each value to the three others, getting four new values. In this case, they would be x^2 + 3x + 5x + 15, or x^2 + 8x + 15.
2b. You can use the Distributive Property to change a quadratic expression from its expanded form into its factores form. A quadratic expression in its expanded form could look like x^2 + 8x + 12. To change it, you have to find out what two numbers, when added, add up to 8, and when multiplied, come out to 12. The two numbers are 2 and 6. When you find these out, the equation will look like (x+2)(x+6).
3. It is very easy to recognize a quadratic function from its equation. All quadratic expressions have an x^2 somewhere in their expanded form. You may mix them up with exponential expressions, so you have to remember, in exponential expressions, the variable is the exponent. In quadratic expressions, the variable is the base.
4. The graphs of quadratic functions are called parabolas. They open either up or down, are symmetrical, and have a vertex. They all have a y-intercept, and have either 2, 1, or 0 x-intercepts. In a quadratic equation, x^2 represents the slope. 2x^2 is steeper than x^2, while 1/3x^2 is less steep. The next part,(in the expanded form), for example, 8x, would represent the two x-intercepts added together. It is much easier to tell what they are in factored form. So, if the equation was (x+4)(x+4), the one x-intercept would be (-4, 0). In this case, the x-intercept is the vertex. The last part(in the equations expanded form), would be 16. This is the y-intercept. For this equation, the y-intercept would be 16.
2a. You can use the Distributive Property to change a quadratic expression from its factored form into its expanded form. A quadratic expression in factored form can look like (x+3)(x+5). To change it to its expanded form, you each value to the three others, getting four new values. In this case, they would be x^2 + 3x + 5x + 15, or x^2 + 8x + 15.
2b. You can use the Distributive Property to change a quadratic expression from its expanded form into its factores form. A quadratic expression in its expanded form could look like x^2 + 8x + 12. To change it, you have to find out what two numbers, when added, add up to 8, and when multiplied, come out to 12. The two numbers are 2 and 6. When you find these out, the equation will look like (x+2)(x+6).
3. It is very easy to recognize a quadratic function from its equation. All quadratic expressions have an x^2 somewhere in their expanded form. You may mix them up with exponential expressions, so you have to remember, in exponential expressions, the variable is the exponent. In quadratic expressions, the variable is the base.
4. The graphs of quadratic functions are called parabolas. They open either up or down, are symmetrical, and have a vertex. They all have a y-intercept, and have either 2, 1, or 0 x-intercepts. In a quadratic equation, x^2 represents the slope. 2x^2 is steeper than x^2, while 1/3x^2 is less steep. The next part,(in the expanded form), for example, 8x, would represent the two x-intercepts added together. It is much easier to tell what they are in factored form. So, if the equation was (x+4)(x+4), the one x-intercept would be (-4, 0). In this case, the x-intercept is the vertex. The last part(in the equations expanded form), would be 16. This is the y-intercept. For this equation, the y-intercept would be 16.
Math Reflection 1
1a. In the graphs of quadratic functions, the line goes up or down in a pattern until it reaches its vertex, when it goes up or down the opposite way. They are symmetrical. They are called parabolas, and they have a y-intercept, and 2 or 0 x-intercepts. In the tables of quadratic functions, the y values go up or down in a pattern, then go up or down the opposite way.
1b. Most of the parabolas weve seen open up, so they go up in a pattern, reach their vertex, and go down in the opposite pattern. In their tables, the y-values go up by the same pattern as the graph, reach the highest number, and go down.
2. One way to find the maximum value for rectangles in a graph is to look at the highest point, or vertex. Another way is to solve the equation l(1/2p-l) by plugging the fixed perimeter and a certain length.
3. The tables of quadratic functions go up by a pattern, reach the vertex(or highesr point), and continue down in the opposite pattern. The tables of exponential functions start at a certain point, then go up by more and more each time. The tables of linear functions go up at a steady rate. The graphs of quadratic functions go up or down in a pattern, reach the vertex, and continue in a way symmetrical to the first half. The graphs of exponential functions start out slowly increasing(or decreasing), then go up very steeply, more so every time. The graphs of linear functions increase at a steady rate. The equations of quadratic functions have a variable multiplied by a quantity with that same variable in it l(1/2p-l). The equations of exponential functions have a certain number raised to a variable, and are in the form of a(b^x). The equations of linear functions are composed of a certain number multiplied by a variable, with another number added on. They are in the form of y=mx+b.
1b. Most of the parabolas weve seen open up, so they go up in a pattern, reach their vertex, and go down in the opposite pattern. In their tables, the y-values go up by the same pattern as the graph, reach the highest number, and go down.
2. One way to find the maximum value for rectangles in a graph is to look at the highest point, or vertex. Another way is to solve the equation l(1/2p-l) by plugging the fixed perimeter and a certain length.
3. The tables of quadratic functions go up by a pattern, reach the vertex(or highesr point), and continue down in the opposite pattern. The tables of exponential functions start at a certain point, then go up by more and more each time. The tables of linear functions go up at a steady rate. The graphs of quadratic functions go up or down in a pattern, reach the vertex, and continue in a way symmetrical to the first half. The graphs of exponential functions start out slowly increasing(or decreasing), then go up very steeply, more so every time. The graphs of linear functions increase at a steady rate. The equations of quadratic functions have a variable multiplied by a quantity with that same variable in it l(1/2p-l). The equations of exponential functions have a certain number raised to a variable, and are in the form of a(b^x). The equations of linear functions are composed of a certain number multiplied by a variable, with another number added on. They are in the form of y=mx+b.
Math Reflection 2
1. The area of a rectangle can illustrate the Distributive Property. For example, if the rectangle is split into four parts, the sides of the original square would both be x and the two additional sides might be 2 and 4. You multiply all the values to each other to find the area of each of the four sections. One would be x^2, one would be 2x, one would be 4x, and the fourth would be 8. You combine the four areas and the equation would be x^2+8x+8.
2a. If a quadratic expression is in factored form, you can change it to expanded form using the Distributive Property. If the equation was (x+3)(x+4), you multiply each term by the other three: x(x), x(3), x(4), 3(4). You get x^2+3x+4x+12, and you combine like terms to get the final equation: x^2+7x+12.
b. If a qaudratic expression is in expanded form, you can change it to factored form using the Distributive Property. If the equation was x^2+11x+30, you have to break up all the terms. x^2 becomes x and x. To figure out the other two terms, you have to find a pair of values that add up to 11 and multiply together to 30. The pair is 5 and 6 because 5+6=11 and 5(6)=30. The new equation is (x+5)(x+6).
3. You can recognize a quadratic function from its equation if it has an exponent that's not a variable (expanded form) or if it has two quantities containing x that are being multiplied (factored form).
4. The graph of a quadratic function is a parabola. The x-intercepts are the opposite of the two numbers in the equation. If the equation is (x+9)(x-2) the x-intercepts are -9 and 2. The y-intercept is the last term in the expanded form equation. If the equation is x^2+5x+6 the y-intercept is 6.
2a. If a quadratic expression is in factored form, you can change it to expanded form using the Distributive Property. If the equation was (x+3)(x+4), you multiply each term by the other three: x(x), x(3), x(4), 3(4). You get x^2+3x+4x+12, and you combine like terms to get the final equation: x^2+7x+12.
b. If a qaudratic expression is in expanded form, you can change it to factored form using the Distributive Property. If the equation was x^2+11x+30, you have to break up all the terms. x^2 becomes x and x. To figure out the other two terms, you have to find a pair of values that add up to 11 and multiply together to 30. The pair is 5 and 6 because 5+6=11 and 5(6)=30. The new equation is (x+5)(x+6).
3. You can recognize a quadratic function from its equation if it has an exponent that's not a variable (expanded form) or if it has two quantities containing x that are being multiplied (factored form).
4. The graph of a quadratic function is a parabola. The x-intercepts are the opposite of the two numbers in the equation. If the equation is (x+9)(x-2) the x-intercepts are -9 and 2. The y-intercept is the last term in the expanded form equation. If the equation is x^2+5x+6 the y-intercept is 6.
Math Reflection 2
1. The area of a rectangle can be illustrated through Distributive Property by breaking up the rectangle into smaller pieces. Say you have a rectangle that has a side of x units, plus another 3 units, and then another side length of x units plus 5 units. In factored form, the equation for getting the area of the rectangle would be (x+3)(x+5). That would mean you would multiply: x by 5, x by x, x by 3, and 3 by 5. You would have to distribute all of these terms to get the area of the rectangle, resulting in the expanded equation of x^2+8x+15.
2a. If an equation is in factored form, you can distribute to get it into expanded form. For example, if an equation were (x+4)(x+4), you would first distribute all the terms: x(x)+4x+4x+4(4). If you simplify that, then the expanded form would be x^2+8x+16.
2b. If an equation is in expanded form, you can do the opposite of distributing and factor to get a factored equation. You can take the above equation, x^2+8x+16, and factor. You need to find 2 numbers that add up to 8, but also multiply together to get 16. Those numbers would be 4 and 4. After finding those numbers, the equation can now be read as x^2+4x+4x+4(4). To factor this, split up the equations into sections. For this particular one, I'll just split it in half: (x^2+4x) and (4x+4(4)). You need to do the opposite of distributing and find out how a number can be multiplied to get what is already there. For the first section, x can be multiplied by x and 4 to get x(x+4). For the second section, 4 can be multiplied by x and 4 to get 4(x+4). To simplify this, the equation would now be factored into x(x+4)+4(x+4) and then further factored into (x+4)(x+4).
3. You can recognize a quadratic function from its equation if the expanded equation as a coefficient of x, raised by a power. In the factored form, you can recognize a quadratic function if the equation involves x being multiplied by x.
4. The shape of a graph of a quadratic equation is that of a parabola.
If the graph has an equation, you can find its y-intercept, x-intercepts, maximum/minimum point, and line of symmetry.
The y-intercept is the number that is the product of the two constants in the equation of the graph.
The x-intercepts are the opposites of those same two constants.
To find the maximum or minimum point , you find the average of the 2 x-intercepts and make that the x-coordinate. To find the remaining y-coordinate, you plug the x-coordinate you got into the equation and solve.
Finally, the line of symmetry of the graph is the x-coordinate of the maximum/minimum point.
2a. If an equation is in factored form, you can distribute to get it into expanded form. For example, if an equation were (x+4)(x+4), you would first distribute all the terms: x(x)+4x+4x+4(4). If you simplify that, then the expanded form would be x^2+8x+16.
2b. If an equation is in expanded form, you can do the opposite of distributing and factor to get a factored equation. You can take the above equation, x^2+8x+16, and factor. You need to find 2 numbers that add up to 8, but also multiply together to get 16. Those numbers would be 4 and 4. After finding those numbers, the equation can now be read as x^2+4x+4x+4(4). To factor this, split up the equations into sections. For this particular one, I'll just split it in half: (x^2+4x) and (4x+4(4)). You need to do the opposite of distributing and find out how a number can be multiplied to get what is already there. For the first section, x can be multiplied by x and 4 to get x(x+4). For the second section, 4 can be multiplied by x and 4 to get 4(x+4). To simplify this, the equation would now be factored into x(x+4)+4(x+4) and then further factored into (x+4)(x+4).
3. You can recognize a quadratic function from its equation if the expanded equation as a coefficient of x, raised by a power. In the factored form, you can recognize a quadratic function if the equation involves x being multiplied by x.
4. The shape of a graph of a quadratic equation is that of a parabola.
If the graph has an equation, you can find its y-intercept, x-intercepts, maximum/minimum point, and line of symmetry.
The y-intercept is the number that is the product of the two constants in the equation of the graph.
The x-intercepts are the opposites of those same two constants.
To find the maximum or minimum point , you find the average of the 2 x-intercepts and make that the x-coordinate. To find the remaining y-coordinate, you plug the x-coordinate you got into the equation and solve.
Finally, the line of symmetry of the graph is the x-coordinate of the maximum/minimum point.
Math Reflection 1
1a. The graphs we've observed have parabolas that open down, and the tables we've observed have values that increase and decrease using the same numbers.
1b. The tables' values increase until it reaches the square of the middle number, and then it decreases back to the number it started with.
2. To find the maximum area for rectangles with a fixed perimeter using a graph, you go to the top of the parabola.
Using a table, you square the middle number.
3. Quadratic graphs have parabolas, while linear graphs have a straight line going at a constant rate and exponential graphs have a curve that's not a parabola.
Quadratic table values increase and decrease, while linear and exponential table values either increase or decrease.
Quadratic equations are modeled like: x(x+b)=y, and have an exponent that does not equal x and is a number larger than 2. Linear equations, on the other hand, are modeled like: y=mx+b, and do not have exponents at all. Finally, exponential equations have an exponent that equals x, and are modeled like: y=a(b)^x
-Sam and Diana
1b. The tables' values increase until it reaches the square of the middle number, and then it decreases back to the number it started with.
2. To find the maximum area for rectangles with a fixed perimeter using a graph, you go to the top of the parabola.
Using a table, you square the middle number.
3. Quadratic graphs have parabolas, while linear graphs have a straight line going at a constant rate and exponential graphs have a curve that's not a parabola.
Quadratic table values increase and decrease, while linear and exponential table values either increase or decrease.
Quadratic equations are modeled like: x(x+b)=y, and have an exponent that does not equal x and is a number larger than 2. Linear equations, on the other hand, are modeled like: y=mx+b, and do not have exponents at all. Finally, exponential equations have an exponent that equals x, and are modeled like: y=a(b)^x
-Sam and Diana
Saturday, May 8, 2010
Math Reflection 2- p.39
1. The area of a rectangle illustrates the Distributive Property because it allows you to see how the different parts of the side legnth affect the overall area. When you have an equation for the area of a rectangle, you often need to distribute to find another form or a value. For example, with the equation A=(x+3)(x+4), you would distribute the x to both the numbers in the second parenthesis, and then the 3, to get the expanded form of the equation.
2.a. If an equation is in factored form, you use the Distributive Property to find the equation in expanded form. For the equation y=(x+2)(x+4) you would distribute the numbers in the first set of parenthesis to both the numbers in the second parenthesis.
b. If an equation is in expanded form and you need to put it in factored form you can use an almost reverse distributive property. If the equation is y=x^2+7x+10, you first find two numbers that add up to 7 and multiply to 10, 5 and 2. You rewrite the equation as y=x^2+5x+2x+10. You then look at the numbers and find, for every set of two numbers, the thing in common. For this equation you have x in common for x^2 and 5x, so you write x(x+5) then you add to that what you get from the other two numbers. In common from 2x and 10, you get 2. So you write, 2(x+5). In both the parts you have something in common, (x+5). You take this as one of the sets in parenthesis, and for the other, you take the things that were multiplied into those parenthesis and get (x+2), so you have y= (x+5)(x+2).
3. You can recognize a quadratic function from its equation if it has the right form, either the expanded form or the factored form.
4. The graph of a quadratic function is a parabola, an symmetrical ark-shape. Some important features on these graphs are the x and y-axis, the minimum\maximum point, and the line of symmetry. You find the x-intercepts by finding the two numbers added to x in the factored form of the equation. You find the y-intercept by if there is a number added\subtracted in the expanded form. You can find the minimum\maximum point by adding together the x-intercepts and dividing by 2 to get the x coordinate. To find the y coordinate you plug the x coordinate into the equation and solve. The line of symmetry is the x-coordinate of the minimum\maximum point.
by Allie :)
2.a. If an equation is in factored form, you use the Distributive Property to find the equation in expanded form. For the equation y=(x+2)(x+4) you would distribute the numbers in the first set of parenthesis to both the numbers in the second parenthesis.
b. If an equation is in expanded form and you need to put it in factored form you can use an almost reverse distributive property. If the equation is y=x^2+7x+10, you first find two numbers that add up to 7 and multiply to 10, 5 and 2. You rewrite the equation as y=x^2+5x+2x+10. You then look at the numbers and find, for every set of two numbers, the thing in common. For this equation you have x in common for x^2 and 5x, so you write x(x+5) then you add to that what you get from the other two numbers. In common from 2x and 10, you get 2. So you write, 2(x+5). In both the parts you have something in common, (x+5). You take this as one of the sets in parenthesis, and for the other, you take the things that were multiplied into those parenthesis and get (x+2), so you have y= (x+5)(x+2).
3. You can recognize a quadratic function from its equation if it has the right form, either the expanded form or the factored form.
4. The graph of a quadratic function is a parabola, an symmetrical ark-shape. Some important features on these graphs are the x and y-axis, the minimum\maximum point, and the line of symmetry. You find the x-intercepts by finding the two numbers added to x in the factored form of the equation. You find the y-intercept by if there is a number added\subtracted in the expanded form. You can find the minimum\maximum point by adding together the x-intercepts and dividing by 2 to get the x coordinate. To find the y coordinate you plug the x coordinate into the equation and solve. The line of symmetry is the x-coordinate of the minimum\maximum point.
by Allie :)
Mathematical Reflection 2
1. The area of a rectangle can illustrate the Distributive Property. For instance, if the lengths of the rectangle were (x+2) by (x+3), the area of the rectangle would become (x+2)(x+3). By drawing it out as a rectangle, we can break up each term separately and distribute, or multiply, them to the others.
2. a) When a quadratic expression is in its factored form, we can use the distributive property to find an equivalent expression in expanded form. If we have an expression, let's suppose (x+2)(x+3), we can distribute each term in the first set of parenthesis into the second, and come up with x(x)+2x+3x+6, or x^2+5x+6.
2. b) To find the equivalent factored form of a quadratic expression written in expanded form, we factor, or the opposite of distributing. Here's an example: The expanded form is x^2+5x+6. Often times we need to break up the second term, which in this case is 5x. We need to find 2 integers that add up to 5, and multiply up to 6. The result is 2 and 3. This expression can now be written as x^2+2x+3x+6. Next, break the expression into two groups, (x^2+2x) and (3x+6). Doing the opposite of distributing, we get x(x+2)+3(x+2). This can be further simplified to (x+2)(x+3), the factored form.
3. A quadratic expression, in its expanded form, follows the form y=ax^2+bx+c.
4. From the equation (x+2)(x+3), we can predict many things. The x-intercepts are the opposite of the integers, or -2 and -3. The y-intercept is those multiplied together, or 6. We can tell if the parabola is pointing up or down by looking at if ax^2 is positive or negative. In this case, it is positive, so it is pointing upwards. The line of symmetry is the x value between the x-intercepts.
Ada =]
2. a) When a quadratic expression is in its factored form, we can use the distributive property to find an equivalent expression in expanded form. If we have an expression, let's suppose (x+2)(x+3), we can distribute each term in the first set of parenthesis into the second, and come up with x(x)+2x+3x+6, or x^2+5x+6.
2. b) To find the equivalent factored form of a quadratic expression written in expanded form, we factor, or the opposite of distributing. Here's an example: The expanded form is x^2+5x+6. Often times we need to break up the second term, which in this case is 5x. We need to find 2 integers that add up to 5, and multiply up to 6. The result is 2 and 3. This expression can now be written as x^2+2x+3x+6. Next, break the expression into two groups, (x^2+2x) and (3x+6). Doing the opposite of distributing, we get x(x+2)+3(x+2). This can be further simplified to (x+2)(x+3), the factored form.
3. A quadratic expression, in its expanded form, follows the form y=ax^2+bx+c.
4. From the equation (x+2)(x+3), we can predict many things. The x-intercepts are the opposite of the integers, or -2 and -3. The y-intercept is those multiplied together, or 6. We can tell if the parabola is pointing up or down by looking at if ax^2 is positive or negative. In this case, it is positive, so it is pointing upwards. The line of symmetry is the x value between the x-intercepts.
Ada =]
Mathematical Reflection 2
1) You can use a rectangle to model the distributive property. The rectangle shown at the bottom of the post illustrates the expression (x+3)(x+4). When simplifying this expression using the distributive property, you would distribute the x and the 3 separately, so it would look like this: x(x+4)+3(x+4). You would solve this by adding up all the products of your distribution. The model shows all the separate products as parts of a whole. You distribute the 3 and x to the x and the 4, which gives you four separate values, which are represented by the four parts of the rectangle. 2)a. To put an expression in factored form into expanded form, you could distribute. You would take one quantity and distribute the value(s) in it separately to the other quantity. For (x+4)(x+2), you would distribute like this: x(x+2)+4(x+2). b. It is generally easier to work with quadratic expressions when they are in factored form, but that means that there will be times when you have to change from expanded to factored. Expanded form usually looks like this: x²+5x+6. There is sometimes more than one way to write the expression in factored form. First, you have to split up the coefficient of x, and the two values have to multiply to the constant in the expression. 3) A function's equation is quadratic if the highest power of the variable is 2 in standard form, and in factored form the equation must have exactly two linear factors to be quadratic. 4) A quadratic function's graph is always a parabola. In factored form, the opposites of the two constants are the x-intercepts, and the line of symmetry is the value between them. The difference between the constants will tell you if it is opening up or down on the coordinate plane. |
Math Reflection p. 39
1. If you have a square, the area is x^2. If you add to the length and width (ex. 3 and 5), you get a new rectangle and therefore a new equation for the area. (x+3)(x+5). The Distributive Property is when you distribute an equation in factored form, (x+3)(x+5) to get a simplified expression in expanded form, x^2+8x+15. The rectangle that has a length of x+3 and a width of x+5 has those two equations: (x+3)(x+5) and x^2+8x+15. You can find the area by using either of these two equations.
2. a) If a quadratic expression is in factored form, you can change it to expanded form by distributing.
factored form: (x+3)(x+5)
You have to multiply the first x by everything in the second set of parenthesis. So you have x*x or x^2 and x*5 or 5x. Now you have to multiply the number in the first set of parenthesis (in this case 3) by everything in the second set of parenthesis. So you have 3*x or 3x and 3*5 or 15. Now your equation in expanded form is x^2+5x+3x+15 or x^2+8x+15.
2. b) If a quadratic expression is in expanded form, you can change it to factored form by splitting each term into smaller terms that will then be able to be distributed back into the same equation in expanded form.
expanded form: x^2+8x+15
The x^2 is easily split into x*x so you know that the factored equation will be (x+?)(x+?). The 8x can be split into 1 and 7, 2 and 6, 3 and 5, or 4 and 4. To know which one it is, you have to figure out the numbers that multiply to 15. 15 can be split into 1 and 15 or 3 and 5. To determine which pair of numbers is correct you have to figure out which one is in both sets. 3 and 5 are in both sets so you know that they will fit into the equation. Therefore, you know what numbers will fit the ?'s. (x+3)(x+5). That is the factored form of the equation x^2+8x+15.
3. You can recognize a quadratic function from its equation by noticing if there is an exponent and if the exponent is a number or a variable. If the exponent is a number then it is a quadratic equation, and if it is a variable then it isn't quadratic. If it is an equation that you can distribute you will have to do that to figure out if there is an exponent and if it is a number or variable.
4. The shape of a quadratic function is always a parabola. Whether it opens up or down, that depends on the equation. A quadratic function usually has two x-intercepts and one y-intercept. To find the x-intercepts, the equation has to be in factored form. You take the opposite of the numbers that are added to x. For example, the equation (x+3)(x+5) would have two x-intercepts, -3 and -5. To find the y-intercept, the equation has to be in expanded form. It is the number added without being multiplied by x. For example, the equation x^2+8x+15 has a y-intercept of 15. Quadratic functions also have a line of symmetry and a maximum or minimum point depending on whether the parabola opens up or down. The equation for the line of symmetry can be found by averaging the two x-intercepts. The x-intercepts of the equation (x+3)(x+5) are -3 and -5 so -3-5=-8/2=-4. Therefore, the line of symmetry is x=-4. To know if the graph has a maximum or minimum point you have to figure out if it opens up or down. The equation x^2+8x+15 has a graph of a parabola opening up. That means it has a minimum point. To find it you can plug the equation for the line of symmetry into the equation to find the y-value of the minimum point. The x-value of the minimum point is the line of symmetry (x=-4 in this case). The line of symmetry is x=-4 so you plug in -4 for the x-values in the equation, x^2+8x+15. -4^2+8(-4)+15=16-32+15=-16+15=-1. Therefore, the minimum point is (-4,-1).
2. a) If a quadratic expression is in factored form, you can change it to expanded form by distributing.
factored form: (x+3)(x+5)
You have to multiply the first x by everything in the second set of parenthesis. So you have x*x or x^2 and x*5 or 5x. Now you have to multiply the number in the first set of parenthesis (in this case 3) by everything in the second set of parenthesis. So you have 3*x or 3x and 3*5 or 15. Now your equation in expanded form is x^2+5x+3x+15 or x^2+8x+15.
2. b) If a quadratic expression is in expanded form, you can change it to factored form by splitting each term into smaller terms that will then be able to be distributed back into the same equation in expanded form.
expanded form: x^2+8x+15
The x^2 is easily split into x*x so you know that the factored equation will be (x+?)(x+?). The 8x can be split into 1 and 7, 2 and 6, 3 and 5, or 4 and 4. To know which one it is, you have to figure out the numbers that multiply to 15. 15 can be split into 1 and 15 or 3 and 5. To determine which pair of numbers is correct you have to figure out which one is in both sets. 3 and 5 are in both sets so you know that they will fit into the equation. Therefore, you know what numbers will fit the ?'s. (x+3)(x+5). That is the factored form of the equation x^2+8x+15.
3. You can recognize a quadratic function from its equation by noticing if there is an exponent and if the exponent is a number or a variable. If the exponent is a number then it is a quadratic equation, and if it is a variable then it isn't quadratic. If it is an equation that you can distribute you will have to do that to figure out if there is an exponent and if it is a number or variable.
4. The shape of a quadratic function is always a parabola. Whether it opens up or down, that depends on the equation. A quadratic function usually has two x-intercepts and one y-intercept. To find the x-intercepts, the equation has to be in factored form. You take the opposite of the numbers that are added to x. For example, the equation (x+3)(x+5) would have two x-intercepts, -3 and -5. To find the y-intercept, the equation has to be in expanded form. It is the number added without being multiplied by x. For example, the equation x^2+8x+15 has a y-intercept of 15. Quadratic functions also have a line of symmetry and a maximum or minimum point depending on whether the parabola opens up or down. The equation for the line of symmetry can be found by averaging the two x-intercepts. The x-intercepts of the equation (x+3)(x+5) are -3 and -5 so -3-5=-8/2=-4. Therefore, the line of symmetry is x=-4. To know if the graph has a maximum or minimum point you have to figure out if it opens up or down. The equation x^2+8x+15 has a graph of a parabola opening up. That means it has a minimum point. To find it you can plug the equation for the line of symmetry into the equation to find the y-value of the minimum point. The x-value of the minimum point is the line of symmetry (x=-4 in this case). The line of symmetry is x=-4 so you plug in -4 for the x-values in the equation, x^2+8x+15. -4^2+8(-4)+15=16-32+15=-16+15=-1. Therefore, the minimum point is (-4,-1).
MATH REFLECTION 2
1. The area of a rectangle demonstrates distributive property be multiplying the base by the height. Suppose you have a rectangle with a hieght of x and a base of x + 7. You gat the equation x(x+7). In order to solve this equation you need to distribute and multiply x by x and x by 7. After solving you get x squared + 7x.
2a. When turning quadratic equations from factored to expanded you can use distributive property. The equation (x+3)(x+5) is in factored form. Now to turn it into expanded you will distribute the (x+3) to (x+5). First multiply x(x+5) and then add 3(x+5). You will get x squared +8x +15.
b. When changing expanded form of quadratic equations to the factored form you can use+distributive property. The normal format for an expanded equation is x squared + yx + z. The two numbers that add up to y must also multiply to equal z. Now knowing this you can change x squared + 5x +4. You must find the two numbers that add up to 5 (y) and that also multiply out to equal 4 (z). Once you have found these you take the two numbers and write the equation again but instead of writing y you write ax + bx (a and b are the two number that add up to y and multiply out to z). x squared + x + 4x + 4. You will then distribute one of the x's in x squared to (x+1) and the 4 in 4x and distribute it to (x+1).
3. You can tell whether and equation is quadratic by looking to see if there are any variables that are squared or if there are any of the same variables being multiplied together.
4. Quadratic graphs are parabolas which are lines that look like arches or U's. There will most likely be two x intercepts if the line crosses the x axis. The x intercepts are given to you in the equation, especially if the equation is in factored form. If your equation is y=(x=5)(x-3) your x intercepts are 5,0 and -3,0. The y intercepts is the number that is directly between the two x intercepts. So in this situation it is 0, 1.
Nic S., Joe P., Mike K., and Harry K.
Friday, May 7, 2010
Math Reflection pg 39
1) The area of a rectangle can illustrate distributive property. In expanded form, the equation of a rectangle is simply the added areas of the smaller rectangles that make up the main rectangle. For example, one of the rectangles we examined had an equation A=x^2+10x+25. The same rectangle had a factored equation that looked like this: A=(x+5)(x+5). These equations show that the same rectangle had both A=(x+5)(x+5) and A=x^2+10x+25 for equations. These are equal because the equation A=(x+5)(x+5) distributed is A=x^2+10x+25. This shows what distribution is, simplifying one equation by distributing terms.
2)a.) When an equation is in factored form you can easily find out it's expanded form. You simply distribute. For example: x(x+5)=x^2+5x because by distributing x you can find x*x=x^2 and x*5=5x.
2)b.) You can find an equations factored form by looking at it's expanded form. What you have to do is find a way to split the equation into two mini-equations in parentheses. to do this you split the "X's" into two parts. When split, the X's need to add up to their original value, and multiply to the y-intercept or the number being added to the X's. here's an example:
x^2+13x+12 this equation need to be split into factored form
the 13 x needs to be split
If you split the 13x into 12x and 1x, you have a sum of 13x and when you multiply 12 and 1 you get 12
So you do x(x+1)+12(x+1) or (x+12)(x+1)
that is equal to the original equation x^2+13x+12.
3) You can recognize a quadratic function by its equation because it has a degree no larger than a two.
4) You can tell the y-intercept, xintercept, line of symmetry and minimum or maximum point by looking at a quadratic function's equations. You can find the y-intercept when the equation is in expanded form. It is the number being added or subtracted from the equation. In the equation y=x^2-25 , the -25 is the y-intercept. In the equation y+2x^2+43, 43 is the y-intercept. You can find the x-intercept when the quadratic expression is in factored form. It is the number you add to the summed parentheses to make it equal 0. Often there are two x-intercepts because the parabola of a quadratic function often intercepts the x-axis twice. Here's an examples of finding x-intercepts.
y=(x+5) (x-4) The x-ints are -5 and 4 because you add 5 to -5 to get 0 and 4 to -4 to get 0. It could also be seen as the opposite of whatever number is being added/subtracted from the x.
To find the line of symmetry you average the two x-intercepts. For example, above the x-ints are -5 and 4. -5+4= -1. -1÷2= -1/2. The line of symmetry is -1/2. Then to find the minimum point you plug -1/2 as x into the equation because the line of symmetry's equation is x=-1/2. So to find the minimum point you do y=(-1/2+5) (-1/2-4)=-20.25. -20.25 is the minimum point.
by Kate :-)
2)a.) When an equation is in factored form you can easily find out it's expanded form. You simply distribute. For example: x(x+5)=x^2+5x because by distributing x you can find x*x=x^2 and x*5=5x.
2)b.) You can find an equations factored form by looking at it's expanded form. What you have to do is find a way to split the equation into two mini-equations in parentheses. to do this you split the "X's" into two parts. When split, the X's need to add up to their original value, and multiply to the y-intercept or the number being added to the X's. here's an example:
x^2+13x+12 this equation need to be split into factored form
the 13 x needs to be split
If you split the 13x into 12x and 1x, you have a sum of 13x and when you multiply 12 and 1 you get 12
So you do x(x+1)+12(x+1) or (x+12)(x+1)
that is equal to the original equation x^2+13x+12.
3) You can recognize a quadratic function by its equation because it has a degree no larger than a two.
4) You can tell the y-intercept, xintercept, line of symmetry and minimum or maximum point by looking at a quadratic function's equations. You can find the y-intercept when the equation is in expanded form. It is the number being added or subtracted from the equation. In the equation y=x^2-25 , the -25 is the y-intercept. In the equation y+2x^2+43, 43 is the y-intercept. You can find the x-intercept when the quadratic expression is in factored form. It is the number you add to the summed parentheses to make it equal 0. Often there are two x-intercepts because the parabola of a quadratic function often intercepts the x-axis twice. Here's an examples of finding x-intercepts.
y=(x+5) (x-4) The x-ints are -5 and 4 because you add 5 to -5 to get 0 and 4 to -4 to get 0. It could also be seen as the opposite of whatever number is being added/subtracted from the x.
To find the line of symmetry you average the two x-intercepts. For example, above the x-ints are -5 and 4. -5+4= -1. -1÷2= -1/2. The line of symmetry is -1/2. Then to find the minimum point you plug -1/2 as x into the equation because the line of symmetry's equation is x=-1/2. So to find the minimum point you do y=(-1/2+5) (-1/2-4)=-20.25. -20.25 is the minimum point.
by Kate :-)
Thursday, May 6, 2010
Math Reflection
1.a. Graphs and tables of quadratic functions both escalate to a maximum and then decrease symmetrically. The greatest area of a rectangle with a fixed perimeter occurs when the length and width are the same, which appears on the graph as the quadratic function's maximum. A quadratic functions graph also has two x-intercepts.
1.b. In a quadratic function, both the tables and graphs escalate to a maximum and then decrease symmetrically. When x is the length of a rectangle and y is the rectangle's area, and the rectangle has a fixed perimeter, the maximum is the rectangle when the rectangle's length and width are the same. This is an example of a quadratic function.
2. One way to find the maximum area for rectangles with a fixed perimeter is to find the highest point on the parabola representing the quadratic function on a graph. You can also find the maximum area with a table. The greatest area is when the length and width are the same.
3. Tables, graphs, and equations of quadratic functions do not increase or decrease at a steady rate like those for linear functions, nor do they curve exactly like exponential functions. Graphs of quadratic functions always show a parabola, which is unique to quadratic functions.
by Noah S., Peter S., and Ryan F.
1.b. In a quadratic function, both the tables and graphs escalate to a maximum and then decrease symmetrically. When x is the length of a rectangle and y is the rectangle's area, and the rectangle has a fixed perimeter, the maximum is the rectangle when the rectangle's length and width are the same. This is an example of a quadratic function.
2. One way to find the maximum area for rectangles with a fixed perimeter is to find the highest point on the parabola representing the quadratic function on a graph. You can also find the maximum area with a table. The greatest area is when the length and width are the same.
3. Tables, graphs, and equations of quadratic functions do not increase or decrease at a steady rate like those for linear functions, nor do they curve exactly like exponential functions. Graphs of quadratic functions always show a parabola, which is unique to quadratic functions.
by Noah S., Peter S., and Ryan F.
Sunday, May 2, 2010
Math Refection
1a) The characteristics of graphs of quadratic functions I have observed, so far, have been parabolas. Of tables of quadratic functions, the area goes up until it gets to where the length and width are the same, and then they start to go back down.
1b) Patterns in a graph of a quadratic function appear in the table of values for the function by going up until they get to where the length and width are the same and then go back down to zero.
2) Two ways to find the maximum area for rectangles with a fixed perimeter are to look at the middle area in a table, and to look at the middle of the parabola graph.
3) Tables, graphs, and equations for quadratic functions are very different from those for linear and exponential functions. Tables are different because they don't go up by a specific number or multiplied by a specific number each time. Graphs are because they are not staright lines and they go up and down, not just up. Equations are different because they are not y=mx+b form or y=m to the x+n.
1b) Patterns in a graph of a quadratic function appear in the table of values for the function by going up until they get to where the length and width are the same and then go back down to zero.
2) Two ways to find the maximum area for rectangles with a fixed perimeter are to look at the middle area in a table, and to look at the middle of the parabola graph.
3) Tables, graphs, and equations for quadratic functions are very different from those for linear and exponential functions. Tables are different because they don't go up by a specific number or multiplied by a specific number each time. Graphs are because they are not staright lines and they go up and down, not just up. Equations are different because they are not y=mx+b form or y=m to the x+n.
Math Reflection 1 Pg. 18
1a. The graphs I have observed have all been parabolas or upside down U's. Also the graph is symmetrical. The highest point on the graph is always the square. The tables show the shapes facing a different direction. The are on the table increase to a certain point then decrease at the same pattern.
1b. The tables display the pattern on the graph by using coordinate points that go together in table form.
2. One way to find the maximum area for a rectangle with a fixed perimeter is to square the number on the x axis below the highest point on the graph. Another way is to find the largest length on the table and square that.
3. The rate is unsteady in quadratics. The graph is a parabola instead of a line. Also it goes up and down in the same equation.
By, Joe P., Nick S., Harry K. and Mike K.
1b. The tables display the pattern on the graph by using coordinate points that go together in table form.
2. One way to find the maximum area for a rectangle with a fixed perimeter is to square the number on the x axis below the highest point on the graph. Another way is to find the largest length on the table and square that.
3. The rate is unsteady in quadratics. The graph is a parabola instead of a line. Also it goes up and down in the same equation.
By, Joe P., Nick S., Harry K. and Mike K.
Friday, April 30, 2010
Math Reflection
1a. All of the graphs have been parabolas and the tables increased in a pattern and decreased in the same pattern after the middle point.
b. With both, it increases in the same pattern and then decreases in that pattern when it reaches the middle point.
2. You can multiply the x-value of the middle point by 4 or double the number in the equation.
3. For tables of quadratic functions, the y-value increases until it reaches the middle point, then decreases in the same pattern. For tables of linear functions, both values increase at a steady rate. For tables of exponential functions, the y-value doesn't increase at a steady rate or with a pattern. The graphs of quadratic functions are parabolas and the middle point is called the vertex. The graphs of linear functions are straight lines. The graphs of exponential functions start at a slow increase and then increase rapidly. The equations all have different parts that make them that certain kind of equation. In a quadratic function, the equation has a variable multiplied by a quantity. A=l(1/2P-l). In a linear function, the equation multiplies two variables and adds another. y=mx+b. In an exponential function, the equation raises a variable to a certain power and multiplies another number to it. y=ab^x.
Emily L. and Alex S.
b. With both, it increases in the same pattern and then decreases in that pattern when it reaches the middle point.
2. You can multiply the x-value of the middle point by 4 or double the number in the equation.
3. For tables of quadratic functions, the y-value increases until it reaches the middle point, then decreases in the same pattern. For tables of linear functions, both values increase at a steady rate. For tables of exponential functions, the y-value doesn't increase at a steady rate or with a pattern. The graphs of quadratic functions are parabolas and the middle point is called the vertex. The graphs of linear functions are straight lines. The graphs of exponential functions start at a slow increase and then increase rapidly. The equations all have different parts that make them that certain kind of equation. In a quadratic function, the equation has a variable multiplied by a quantity. A=l(1/2P-l). In a linear function, the equation multiplies two variables and adds another. y=mx+b. In an exponential function, the equation raises a variable to a certain power and multiplies another number to it. y=ab^x.
Emily L. and Alex S.
Thursday, April 29, 2010
Math Reflection...to be corrected if wrong!
Lately in Algebra 1, we have been observing the graphs and tables of quadratic functions. The examples we have been using revolve around the relationship between rectangles lengths and areas.
1.a) In the tables we have been viewing, as the length increases, the area increases. This continues until the table reaches the half-way point within the set of length values. At this point the area is the greatest it has been in the table, and then begins to decrease in the opposite way that it increased. The graphs we have observed have all been parabolas that open down. The arch shaped lines The line increases at a curve, and then comes to a summit where the area is the largest, and then decreases at a curve symmetrical to the increasing curve.
1.b)The patterns in a quadratic functions graph also appear in a table. They both increase with the length until the graph/table reaches a maximum area and they decrease in the opposite order that the numbers increased.
2.)One way to find the maximum area of a rectangle with a fixed perimeter is to make a graph or table. The highest point on the graph is the largest area and the x-axis value below that point would be the length that corresponds with the area. Another way to find the maximum area is to square the length of the rectangle. A rectangles length squared is always the maximum area.
3.) The graphs, tables and equations of quadratic functions are different from those of linear and exponential functions. The graph of a quadratic functions are different from linear graphs because they are not straight lines. They are not like exponential functions because they both increase and decrease in one graph. They increase and decrease with a pattern, and end up looking like an arch not a line or a increasing or decreasing curved line. The table of e quadratic function is different from one of a linear function. Unlike a table of e linear function, a table of a quadratic function doesn't increase of decrease at a fixed rate. A table of a quadratic function doesn't multiply of divide itself like one of an exponential function does. The equation of a quadratic function is not like one of a linear function be. this is because it includes exponents which a linear function doesn't. ( for example like finding the width using an equation like 30l-l^2) It isn't like one exponential because in an exponential function the variable is the exponent. But, in a quadratic function the variable is the number that the length is being subtracted by. ( 1/2 of the perimeter). Instead of the exponent changing from equation to equation, in a quadratic function the fixed perimeter of the rectangle is the variable....I think. :-)
by Kate M. and Allie G.
1.a) In the tables we have been viewing, as the length increases, the area increases. This continues until the table reaches the half-way point within the set of length values. At this point the area is the greatest it has been in the table, and then begins to decrease in the opposite way that it increased. The graphs we have observed have all been parabolas that open down. The arch shaped lines The line increases at a curve, and then comes to a summit where the area is the largest, and then decreases at a curve symmetrical to the increasing curve.
1.b)The patterns in a quadratic functions graph also appear in a table. They both increase with the length until the graph/table reaches a maximum area and they decrease in the opposite order that the numbers increased.
2.)One way to find the maximum area of a rectangle with a fixed perimeter is to make a graph or table. The highest point on the graph is the largest area and the x-axis value below that point would be the length that corresponds with the area. Another way to find the maximum area is to square the length of the rectangle. A rectangles length squared is always the maximum area.
3.) The graphs, tables and equations of quadratic functions are different from those of linear and exponential functions. The graph of a quadratic functions are different from linear graphs because they are not straight lines. They are not like exponential functions because they both increase and decrease in one graph. They increase and decrease with a pattern, and end up looking like an arch not a line or a increasing or decreasing curved line. The table of e quadratic function is different from one of a linear function. Unlike a table of e linear function, a table of a quadratic function doesn't increase of decrease at a fixed rate. A table of a quadratic function doesn't multiply of divide itself like one of an exponential function does. The equation of a quadratic function is not like one of a linear function be. this is because it includes exponents which a linear function doesn't. ( for example like finding the width using an equation like 30l-l^2) It isn't like one exponential because in an exponential function the variable is the exponent. But, in a quadratic function the variable is the number that the length is being subtracted by. ( 1/2 of the perimeter). Instead of the exponent changing from equation to equation, in a quadratic function the fixed perimeter of the rectangle is the variable....I think. :-)
by Kate M. and Allie G.
Wednesday, April 14, 2010
Guidelines for Posting Comments
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Getting a comment can be like receiving a little bouquet in your mailbox: a treat for the senses.
Guidelines
• Make your comment worth reading.
• Start a conversation.
• Be positive, interested, and encouraging.
• If you disagree, be polite about it.
• Connect with the post: be on topic.
• Re-read your comment before you hit submit–think before you send!
• Aim for correct spelling, punctuation, and grammar.
• Don’t use chat or texting language like lol, i, or u.
• No “Hi! Nice Job! Bye!” comments. Be thoughtful.
• Keep your privacy: no personal or identifying information about you, your family, or your friends. Don’t give out last names, school name, phone numbers, user names, or places and dates you can be found.
Wednesday, March 31, 2010
Tuesday, March 30, 2010
Math Reflection
1) You can recognize an exponential decay pattern from a table of data if it is divided my the same number each time.
2) You can recognize an exponential decay pattern from a graph when the line slopes down.
3) You can tell that an equation represents exponential decay if the growth factor is less than 1.
4) Exponential growth relationships and exponential decay relationships are similar because they both use exponents, they have growth/decay rates, and they both have growth/decay factors. They are different because exponential growth relationships have a growth factor greater than 1, when an exponential decay's is less than 1.
5) Exponential decay relationships and decreasing linear relationships are similar because they both decrease. They are different because exponential decay relationships are divided by the same number each time, while decreasing linear relationships get the same number smaller each time.
2) You can recognize an exponential decay pattern from a graph when the line slopes down.
3) You can tell that an equation represents exponential decay if the growth factor is less than 1.
4) Exponential growth relationships and exponential decay relationships are similar because they both use exponents, they have growth/decay rates, and they both have growth/decay factors. They are different because exponential growth relationships have a growth factor greater than 1, when an exponential decay's is less than 1.
5) Exponential decay relationships and decreasing linear relationships are similar because they both decrease. They are different because exponential decay relationships are divided by the same number each time, while decreasing linear relationships get the same number smaller each time.
Monday, March 29, 2010
Math Reflection 3
1.
a. If you know the initial value for a population, and you know the yearly growth rate, you can make an equation. For example, starting population=500, yearly growth rate=80%. To change the growth rate to the growth factor, you change the percent to a decimal, and add 1. The equation for this situation would be 500(1.8^x). If you want to find out the population after, say, 5 years, just plug in 5 for x, 500(1.8^5). The answer is 9447.84.
b. The growth rate of an equation is the growth factor of an equation, times 100 (as a percent). If the growth factor was 1.6, the growth rate would be 160%.
2.
a. If you know the initial value of a population and the yearly growth rate, you can make an equation. If the beginning population was 200, and yearly growth factor was 1.3, the equation would be 200(1.3^x). If you wanted to know the population 7 years from now, you would just plug in 7 for the x. The answer is 200(1.3^7), or 1254.97034.
b. You can determine the yearly growth rate by multiplying the growth factor by 100.
3. To get the population when it doubles, just multiply the starting value by 2. Then, guess and check what the x value is(the year), when the population doubles. When it is about the starting value times 2, that is it.
a. If you know the initial value for a population, and you know the yearly growth rate, you can make an equation. For example, starting population=500, yearly growth rate=80%. To change the growth rate to the growth factor, you change the percent to a decimal, and add 1. The equation for this situation would be 500(1.8^x). If you want to find out the population after, say, 5 years, just plug in 5 for x, 500(1.8^5). The answer is 9447.84.
b. The growth rate of an equation is the growth factor of an equation, times 100 (as a percent). If the growth factor was 1.6, the growth rate would be 160%.
2.
a. If you know the initial value of a population and the yearly growth rate, you can make an equation. If the beginning population was 200, and yearly growth factor was 1.3, the equation would be 200(1.3^x). If you wanted to know the population 7 years from now, you would just plug in 7 for the x. The answer is 200(1.3^7), or 1254.97034.
b. You can determine the yearly growth rate by multiplying the growth factor by 100.
3. To get the population when it doubles, just multiply the starting value by 2. Then, guess and check what the x value is(the year), when the population doubles. When it is about the starting value times 2, that is it.
Math Reflection 3
1. A) If you know the initial value and the growth rate of the population you can make an equation using the format y=a(b)^n. Say the initial value was 1000 and the growth rate was 4%, you first need to make the growth factor. You do this by making the percent into a decimal, (.04), then you add 1, (1.04), your equation would be y=1000(1.04)^n. to figure out the population in a certain amount of years you just put the number in as x and solve.
1. B) The growth rate is related to the growth factor because the growth factor is the rate in decimal form plus 1.
2. A) You do the same as above. You take the initial value and growth factor and put them into an equation. Then you put the number of years s x then solve.
2. B) You can determine the growth rate by first subtracting 1. Then you multiply the decimal by 100 to get the percent.
3. To find the doubling point of the population you take the initial value and double it. So now the equation is y=2p(b)^n. From there you take an educated guess to see it when you plug x in (number of years) it doubles. If it doesn't you go up or down from there to see when the doubling point is.
Math Reflection
1. a) To determine the population many years from now you have to figure out the growth factor. To do this, you change the percent to a decimal and add one. For example, if the growth rate is 20%, the growth factor is 1.2. You would then take the growth factor and the initial value and put it into this equation: y=a(b)^x.
y=total, a=initial value, b=growth factor, x=number of years.
If the initial value is 200 and the growth factor was 1.2 the equation would be y=200(1.2)^x.
b) The growth rate is related to the growth factor because they are the same thing just written differently. To change the growth rate (which is a percent) to the growth factor (a number...usually a decimal) you divide by 100 and add 1. To change the growth factor to the growth rate you subtract 1 and multiply by 100. (It's doing the opposite of changing the growth rate to the growth factor).
2. a) If you know the initial value and the growth factor you can just plug it into this equation to find out the total for any number of years. The equation is: y=a(b)^x.
y=total, a=initial value, b=growth factor, x=number of years.
b) To determine the yearly growth rate you subtract 1 from the growth factor and multiply by 100.
3. If you know the equation that represents the exponential relationship between the population and the number of years you can determine the doubling time for the population by first figuring out 2p. After that you plug that value into the equation p=a(b)^x to solve for x. The x will give you how long it will take for the population to double.
Math Reflection from Page 47
#1 When you know the initial value and yearly growth rate, it is simple to determine a population from year to tear. First, convert the growth rate to a growth factor by putting the growth factor into decimal form and adding 1. For example if the growth rate is 7% then 7% in decimal form is 0.07 and 0.07+1=1.07. The growth factor is 1.07. Next, start the equation with the initial population. Then multiply that with the growth rate to the power of the year. For example, if the initial population is 1,000 and the growth factor is 1.07, then the population in 4 years corresponds to the equation 1000(1.07)^4. The population is about 1310 in 4 years. Growth factor and growth rate are closely related. Growth rate is really growth factor but in percent form without including the current sum. Growth factor is growth rate plus one and in the decimal form.
#2 If you know both the initial value and the growth factor, finding a population in a few years is even simpler. All you need to do is multiply the initial value times the growth rate to the power of the year. If the initial population is 10,000 and the growth rate is 1.1 the population is 6 years is found with the following equation: 10,000(1.1)^6.
#3 To determine the doubling population's sum, you use the following equation p= 2^n. This means population equals the growth factor (2) to the power of the year, n.
Kate :-)
#2 If you know both the initial value and the growth factor, finding a population in a few years is even simpler. All you need to do is multiply the initial value times the growth rate to the power of the year. If the initial population is 10,000 and the growth rate is 1.1 the population is 6 years is found with the following equation: 10,000(1.1)^6.
#3 To determine the doubling population's sum, you use the following equation p= 2^n. This means population equals the growth factor (2) to the power of the year, n.
Kate :-)
Math Reflection 3
1.a. You could make an equation by substituting the growth rate(as a decimal and added to 1) as b and the initial value as a.You then substitute the number of years as the x.
1.b.The growth rate is what percent of the original is added to itself. The growth factor is the original value added to what ever is added to the original value.
2.a.You could make an equation using a=initial value b=growth factor. You could then substitute the number of years for x.
2.b. You make the growth factor a percent value and subtract 100% from it.
3. You figure out the amount of years it takes for the growth factor to equal 2 using guess and check.
1.b.The growth rate is what percent of the original is added to itself. The growth factor is the original value added to what ever is added to the original value.
2.a.You could make an equation using a=initial value b=growth factor. You could then substitute the number of years for x.
2.b. You make the growth factor a percent value and subtract 100% from it.
3. You figure out the amount of years it takes for the growth factor to equal 2 using guess and check.
Math Reflection 3
1a. To determine the population several years from the start using the initial value and the yearly growth rate, you can form an equation. To do this, you use the form of y=a(b^x) (a=initial value, b=growth factor, y=dependent variable, x=independent variable). Also, it's important that you take the growth rate, and convert it into its growth factor by taking the percent and adding 1 to its decimal form.
1b. A growth rate is the percent form of a growth factor. For instance, if a growth rate were 75%, its growth factor would be 1.75 because you have to take in account 100% of the previous value, plus the 75% that is going to be added to form the next value of y.
2a. To determine the population several years from the start using initial value and the yearly growth factor, you will also need to form an equation, only this time, there is no need to turn the growth rate into a growth factor, because there is no growth rate at all. Yet again, you use the form of y=a(b^x).
2b. You can determine the yearly growth rate by subtracting 1 from the number that is the growth factor and turning it into a percent.
3. If you know the equation that represents the exponential relationship between the population size p and the number of years n, you can determine the doubling time for the population by finding what 2p is, then finding the value of n that is closest to that amount.
1b. A growth rate is the percent form of a growth factor. For instance, if a growth rate were 75%, its growth factor would be 1.75 because you have to take in account 100% of the previous value, plus the 75% that is going to be added to form the next value of y.
2a. To determine the population several years from the start using initial value and the yearly growth factor, you will also need to form an equation, only this time, there is no need to turn the growth rate into a growth factor, because there is no growth rate at all. Yet again, you use the form of y=a(b^x).
2b. You can determine the yearly growth rate by subtracting 1 from the number that is the growth factor and turning it into a percent.
3. If you know the equation that represents the exponential relationship between the population size p and the number of years n, you can determine the doubling time for the population by finding what 2p is, then finding the value of n that is closest to that amount.
Math Reflection 3
1.a. With the initial value and yearly growth rate, you can determine the x value several years from now by using the equation y=a(b^x) where a is the initial value, b is the yearly growth factor and x is the given value of x. But, before you write the equation yo must turn the growth rate into a growth factor by adding one to the growth rates decimal value.
b. To turn the growth rate into the growth factor, you add one to the growth rate's decimal value. The growth rate is the percent of growth and the growth factor is what the x value is multiplied each time it goes up.
2.a. If you know the initial value and the growth factor, you simply place the numbers in the equations. Substitute the initial value for a, and the growth factor for b. Then you subsitute the x value for x.
b. To find the growth rate out of the growth factor, you subtract one and turn the decimal into a percent.
3. To find the doubling time, you guess and check until it is just about correct.
mathematical reflection 3
1. a. If you know the initial value and the growth rate for a population you can find the population years from now. You can do this by making the growth rate into the growth factor by making it a decimal and adding 1. You then find the growth factor to the how many years it has been power. Then you multiply the initial value by the total of the exponent problem.
1. b. The growth factor is the percent of growth and the growth rate is how much the y total is multiplied by each time x goes up. You can find the growth factor from the growth rate by putting the growth rate into decimal form and adding 1 to the decimal.
2. a. If you know the growth factor and the initial value for a population equation you can find the population several years from now. You can do this by finding the growth to the exponent of how many years it has been and multipying the initial value by the total of the exponent problem.
2. b. You can find the yearly growth rate by using the growth factor. You can do this by subtracting 1 from the growth factor and making the number you have left a percent.
3. You can find the doubling of a population p for the number of years n by making the equation p=2^ n and multiply that by the intial value if there is an initial value.
1. b. The growth factor is the percent of growth and the growth rate is how much the y total is multiplied by each time x goes up. You can find the growth factor from the growth rate by putting the growth rate into decimal form and adding 1 to the decimal.
2. a. If you know the growth factor and the initial value for a population equation you can find the population several years from now. You can do this by finding the growth to the exponent of how many years it has been and multipying the initial value by the total of the exponent problem.
2. b. You can find the yearly growth rate by using the growth factor. You can do this by subtracting 1 from the growth factor and making the number you have left a percent.
3. You can find the doubling of a population p for the number of years n by making the equation p=2^ n and multiply that by the intial value if there is an initial value.
Mathematical Reflections 3
1. a. If your know the initial value for a population and the yearly growth rate, you can determine the population several years from now. First ou find the growth factor, because the growth factor and the initial value are the two things you need to form an equation. To get the growth factor, you put the growth rate into decimal form, and add one. Then you form an equation using the y=a(b^x) form. You plug in the initial value as A and the growth factor as B. Then you can plug the number of years into the equation to find your answer.
b. The growth rate is the percentage of growth, while the growth factor is the amount of increase. You can find the growth factor from the growth rate by putting the percent into decimal form and adding 1.
2. a. If you know the initial value and the yearly growth factor you can determine the population in several years. To do this, you need to form an equation. You can plug th numbers you have into the form y=a(b^x). A is the initial value, and B is the growth factor. Solve the equation, and you have your answer.
b. You can determine the growth rate by subtracting 1 from the growth factor and turning that number into a percent.
3. You can determine the doubling time by making a table and observing when a number doubles.
b. The growth rate is the percentage of growth, while the growth factor is the amount of increase. You can find the growth factor from the growth rate by putting the percent into decimal form and adding 1.
2. a. If you know the initial value and the yearly growth factor you can determine the population in several years. To do this, you need to form an equation. You can plug th numbers you have into the form y=a(b^x). A is the initial value, and B is the growth factor. Solve the equation, and you have your answer.
b. You can determine the growth rate by subtracting 1 from the growth factor and turning that number into a percent.
3. You can determine the doubling time by making a table and observing when a number doubles.
Mathematical Reflection 3
1a. If we know the initial value for a population and the yearly growth rate we can determine the population any number of years from now. We can find the growth factor from the growth rate by adding 1, then writing in decimal form. After that, plug the values of a(the initial value), b(the growth factor), and x(the number of years) into the equation y=a(b^x), and solve for y.
1b. A growth rate is the percentage growth, while the growth factor is the fractional pattern of increase. We find the growth rate by adding 1 to the growth rate and then dividing by 100 to get the decimal form, or growth factor.
2a. In the equation y=a(b^x), we already have the values of a(the initial value) and b(the yearly growth factor). If we substitute the number of years in for x, we automatically can solve for y.
2b. We can find the yearly growth rate from the yearly growth factor by subtracting 1and writing it in a percentage form.
3. To find the doubling time for the population, find the value of 2p. Then, plug the new value of p into the equation to solve for n.
1b. A growth rate is the percentage growth, while the growth factor is the fractional pattern of increase. We find the growth rate by adding 1 to the growth rate and then dividing by 100 to get the decimal form, or growth factor.
2a. In the equation y=a(b^x), we already have the values of a(the initial value) and b(the yearly growth factor). If we substitute the number of years in for x, we automatically can solve for y.
2b. We can find the yearly growth rate from the yearly growth factor by subtracting 1and writing it in a percentage form.
3. To find the doubling time for the population, find the value of 2p. Then, plug the new value of p into the equation to solve for n.
Math Reflection 3
1a. To determine the population several years after and you have the growth rate and initial population you turn the growth rate into a growth factor then make an equation Y= initial value(growth factor to the x).
1b. The growth rate is the percent that the number is going up by and the growth factor is the number that you put to the xth power.
2a. You can once again make the equation and solve. It will be one less step from 1 because you do not have to change the percent to a decimal and then solve.
2b. You turn the number to a percent then subtract 100.
3. To find the time it will take an initial value to double you must find what x is so that the growth factor can equal 2 so it can double. Another way you could do it is by guessing and checking, you geuss for a number to be x and put the growth factor if it is over 2 it can work and you can try a smaller number.
1b. The growth rate is the percent that the number is going up by and the growth factor is the number that you put to the xth power.
2a. You can once again make the equation and solve. It will be one less step from 1 because you do not have to change the percent to a decimal and then solve.
2b. You turn the number to a percent then subtract 100.
3. To find the time it will take an initial value to double you must find what x is so that the growth factor can equal 2 so it can double. Another way you could do it is by guessing and checking, you geuss for a number to be x and put the growth factor if it is over 2 it can work and you can try a smaller number.
Math Reflection
MATH REFLECTION
1a. Provided you were knew the initial population and growth rate, finding the population x years is really quite simple. All you need to do is turn the growth rate into a growth factor. You do this by turning the percentage given to you as a growth rate and change it to a decimal or integer. You will then add one to the number you get from the growth rate. This number is a vital part of this process, as it will be the base of the exponent in the equation. You would then right an equation for maximum clarity. EXAMPLE:
Initial Population: 100
Growth Rate: 5%
Growth Factor: 1.05
Equation: Y=100(1.05 to the power of x)
You now choose the number of years and sub that in for x. After solving you should know the population.
1b. Growth rate and growth factor are very closely related. The growth rate is a percent and the factor a number. The growth rate shows the rate at which each number changes in the equation. While the growth factor is the number that "physically" represents the rate in the equation and is the base of the exponent in the equation.
2a. If you had the initial population and the growth factor of a situation finding the population after a set number of years would be simple. All you would need would be to right an equation. This is easy because you are provided with all the needed elements of the equation. You would make the population (a variable) equal the initial value multiplied by the growth rate to the power of a variable representing the number of years that have passed.
EXAMPLE
Initial Value: 24
Growth Factor: 1.24
Equation: Y = 24(1.24 to the power of x)
2b. When given the growth factor finding the growth rate is not hard. You turn the growth factor into a percentage and then subtract 100%. You can also subtract one from the factor and then turn in into a percent.
3. In order to determine the doubling point of a population when given the equation of that situation you would need to find the exponent that will bring the growth factor to two. This will then double the initial population. Finding the exponent that doubles the factor is purely educated guess and check.
Nick S.
Sunday, March 28, 2010
Mathematical Reflections 3
1)
a. You can determine the population in several years from now by making a equation out of what you know. The formula is y=a(b^x). Since you know the initial value, plug it in to the "a". Then change the growth rate to a growth factor. You can do this by dividing one hundred and adding 1. Then, plug it into "b". Then, you can plug in the amount of years that you want to solve for in the "x", and solve the equation.
b. The growth rate is the percentage of growth between each year in this case. To find the growth rate from a growth factor is to subtract the 1 and multiply by 100. To find the growth factor from a growth rate is to add the 1 and divide by 100. The reason that the growth rate doesn't have the 1, and the growth factor does is because the growth factor is already the original value and the growth. The growth rate is only the growth and not the original value.
2)
a. As I said before, you could use the equation, y=a(b^x). Plug the initial value into "a". Then, as a step less from question 1a, you can just plug in the growth factor. Then, plug in the amount of years that you want to solve for in the "x" then solve.
b. You can determine the yearly growth rate by the growth factor. Subtract the 1 from the growth factor and multiply by 100.
3) You could do two things to determine the doubling time for the population. You could plug in twice the amount of the initial value in to "y", to make it equal that number. Then, you could solve for "x". Or you can do the guess and check method that I usually use. Plug in twice the initial value for "y", but guess a number for "x" and work your way around that to determine the correct answer.
a. You can determine the population in several years from now by making a equation out of what you know. The formula is y=a(b^x). Since you know the initial value, plug it in to the "a". Then change the growth rate to a growth factor. You can do this by dividing one hundred and adding 1. Then, plug it into "b". Then, you can plug in the amount of years that you want to solve for in the "x", and solve the equation.
b. The growth rate is the percentage of growth between each year in this case. To find the growth rate from a growth factor is to subtract the 1 and multiply by 100. To find the growth factor from a growth rate is to add the 1 and divide by 100. The reason that the growth rate doesn't have the 1, and the growth factor does is because the growth factor is already the original value and the growth. The growth rate is only the growth and not the original value.
2)
a. As I said before, you could use the equation, y=a(b^x). Plug the initial value into "a". Then, as a step less from question 1a, you can just plug in the growth factor. Then, plug in the amount of years that you want to solve for in the "x" then solve.
b. You can determine the yearly growth rate by the growth factor. Subtract the 1 from the growth factor and multiply by 100.
3) You could do two things to determine the doubling time for the population. You could plug in twice the amount of the initial value in to "y", to make it equal that number. Then, you could solve for "x". Or you can do the guess and check method that I usually use. Plug in twice the initial value for "y", but guess a number for "x" and work your way around that to determine the correct answer.
Mathematical Reflections 3
1.a. If the initial value and growth rate of a population are known, you can easily determine the population several years from now. First, you must change the growth rate into the growth factor. The growth factor is usually in decimal form, but most important is that for exponential growth it is at least 1 or greater. Then, you just multiply the initial value by the growth factor raised to the n power.
b. The growth rate represents by how much the initial value increases each year, and the growth factor is what it is multiplied by each year. For example, the growth factor would be 1.7 when the growth rate is 70%. (The growth factor must have a 1 during exponential growth in order to make sure that the initial value is kept along with the increase.)
2.a. To find the population from the initial value and growth factor, you would again use the y=a(b)^x form, except this time you don't need to convert anything.
b. To determine the yearly growth rate, you subtract 1 from the growth factor if it is less than 2.
3. To find out how long it will take for a population to double, all you have to do is make y twice the initial value. Another thing you could do is to keep multiplying by the growth factor until the outcome is double or more the initial value.
b. The growth rate represents by how much the initial value increases each year, and the growth factor is what it is multiplied by each year. For example, the growth factor would be 1.7 when the growth rate is 70%. (The growth factor must have a 1 during exponential growth in order to make sure that the initial value is kept along with the increase.)
2.a. To find the population from the initial value and growth factor, you would again use the y=a(b)^x form, except this time you don't need to convert anything.
b. To determine the yearly growth rate, you subtract 1 from the growth factor if it is less than 2.
3. To find out how long it will take for a population to double, all you have to do is make y twice the initial value. Another thing you could do is to keep multiplying by the growth factor until the outcome is double or more the initial value.
Saturday, March 27, 2010
Math Reflections 3
1a. You can determine the population several years from now by changing the growth rate to a growth factor by dividing by 100 and adding 1. Then you write an exponential equation using the initial population and the growth factor and then substitute the number of years for x.
b. A growth rate is related to the growth factor of a population because they both represent the same thing, except a growth rate is in percent form and the growth factor is in decimal form.
2a. You can determine the population several years from now by writing an exponential equation in y=a(b)^x form where y=the population, a=the initial population, b=the growth factor, and x=the number of years. Then, you solve the equation by sunstituting the number of years for x.
b. You can determine the yearly growth rate by subtracting 1 from the growth factor and changing it to a percent.
3. You can determine the doubling time for the population by substituting twice the initial value for y and solving the equation. You could also guess and check until you find the right number of years.
b. A growth rate is related to the growth factor of a population because they both represent the same thing, except a growth rate is in percent form and the growth factor is in decimal form.
2a. You can determine the population several years from now by writing an exponential equation in y=a(b)^x form where y=the population, a=the initial population, b=the growth factor, and x=the number of years. Then, you solve the equation by sunstituting the number of years for x.
b. You can determine the yearly growth rate by subtracting 1 from the growth factor and changing it to a percent.
3. You can determine the doubling time for the population by substituting twice the initial value for y and solving the equation. You could also guess and check until you find the right number of years.
Math Reflections 3
1.a. You can determine the population several years from now, if you know the initial value in yearly growth rate, by changing the growth rate from a percent to a decimal, and by adding 1 to that decimal. Then you can multiply the initial value by this growth factor as many years in the future as you want to determine the population.
1.b. A growth factor is just the decimal conversion of a growth rate plus 1.
2.a. You can find the population several years from now, if you know the initial value and yearly growth factor, by multiplying the initial value by the yearly growth factor as many times as the number of years from now you want to determine the population.
2.b. You can determine the yearly growth rate by subtracting 1 from the growth factor and converting the value you are left with to a percent.
3. You can find the doubling time for a population from an equation by trying different values for the time variable until the population doubles.
1.b. A growth factor is just the decimal conversion of a growth rate plus 1.
2.a. You can find the population several years from now, if you know the initial value and yearly growth factor, by multiplying the initial value by the yearly growth factor as many times as the number of years from now you want to determine the population.
2.b. You can determine the yearly growth rate by subtracting 1 from the growth factor and converting the value you are left with to a percent.
3. You can find the doubling time for a population from an equation by trying different values for the time variable until the population doubles.
Exponential Decay
Exponential decay is when an equation decreases exponentially. Exponential decay equations have a decay factor instead of a growth factor. For example, a chart with exponential decay might look like this:
X / 1 / 2 / 3 /4/5/6
Y/64/32/16/8/4/2
The equation for this chart would be Y= 64(1/2)^x. Whenever you find the numbers are getting smaller each time, you know that there is a decay factor. The decay factor in this equation is 1/2. The decay factor will always be a decimal/fraction. The initial value in this equation is 64.
X / 1 / 2 / 3 /4/5/6
Y/64/32/16/8/4/2
The equation for this chart would be Y= 64(1/2)^x. Whenever you find the numbers are getting smaller each time, you know that there is a decay factor. The decay factor in this equation is 1/2. The decay factor will always be a decimal/fraction. The initial value in this equation is 64.
Thursday, March 25, 2010
Wednesday, March 24,2010
Yesterday in Algebra 1, we worked on connecting grow factor with growth rate. The growth factor is the number in an exponential function which the previous result is multiplied by in order to get the result for the next interval of the function. For example, in the equation y=100(2.2)* 2.2 is the growth factor because it is the number that *(x) is multiplied by to get the next sum in the function. Once we learned about growth factor, we learned about growth rate. The growth rate is the percent of increase in an exponential function, and it is sometimes given in place of the growth factor. When the population is growing 3% a year, 3% is the growth factor. Growth factor and growth rate are easily interchangeable. When the growth factor is 1.23, the growth rate is 23%. This is because when changing growth rate to growth factor you simply put the percentage in decimal form and add one to it (because you add the current amount to the growth rate when you are solving a function). When the money within a bank account is 4,000, and grows 7% a year, all you have to do to find the amount in it after the first year is do 4000 x 1.07. 7% is really 0.07, and once you add one to include the 4000, you get 1.07 as the growth factor. Switching from the growth factor to the growth rate is just as easy as the oppisite. Subtracting 1 from the growth factor and putting the remaining decimal into a percent is all it consists of. When the growth factor is 1.80, you do 1.80-1=.80 .80=80% and you find that the growth rate is eighty percent when the grwoth factor is 1.80. Growth factor and growth rate are simply connected. By Kate M :-)
Wednesday, March 24, 2010
Math Reflection
1.
a) From a table you can find the y-intercept because when x equals zero it's the y-intercept. You can find the growth factor by dividing one number by the number before it on the graph. Then use the answer to see if it works as the growth factor for the rest of the numbers.
From a graph you can find the y-intercept just by looking where the line crosses the y-axis. You can find the growth factor by seeing how many spaces go up each time it goes up by one.
From and equation you can find the y-intercept by looking at the a in y=a(b)^n. You can find the growth factor by looking at the b.
b) Use the y-intercept as the a and use the growth factor as the b.
2.
a) A represents the y-intercept except when c changes it. B represents the growth factor.
Monday, March 22, 2010
Mathematical Reflections 2
1)
a. You can use a table to find the y-intercept by looking at the y when x is 0. You can find the growth factor from the table by dividing two x's and two y's right in above each other. Then, the division of y over x is the growth factor. On a graph, you basically do the same as a table. Find two points preferably right next to each other. Then the division of the y's over the x's is the growth factor. The y-intercept is found when the x coordinate is 0. The initial equation is y=a(b^x). The a is the y-intercept which is just plainly stated. The b is the growth factor which is also plainly stated.
b. As I said in letter 1a. The variable a is the y-intercept. If you know the y-intercept you can plug it in. For example, let's say 2 is the y-intercept. This is how much of the equation you would have. y=2(b^x). B is the growth factor, and same as before just plug it into b. Let's say 3 is the growth factor. The equation now would be y=2(3^x). Now the equation is complete.
2)
a. In the equation y=a(b^x), the a is the initial amount or y-intercept. The b is growth factor.
b. The y-intercept or a is represents how up or down the curved line is.
c. The growth factor or b is represented the steepness of the curved line.
Mathematical Reflection 2
1. a. You can use a table, a graph, and an equation to find the y-intercept and growth factor for an exponential relationship by looking at the table and graph then finding what y is when x is zero
(y-intercept), and what the number multiplies by each time (growth factor).
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship by using the equation y=a(b^x). A= Y-Intercept B= Growth Factor
2. a. In the equation y=a(b^x) a equals the y-intercept and b equals the growth factor.
2. b. A is represented in a graph of y=a(b^x) byecause it is the y-intercept, so a is what y equals when x equals zero.
2. c. B is represented in a graph of y=a(b^x) by making the line steeper or less steep.
Math Reflections 2
1.a. You can use a table to find the y-intercept of an exponential relationship by finding the dependent variable's value when the independent variable's value is 0. When finding the y-intercept of an exponential relationship in a graph, you look to see where the line crosses the y-axis. To find the y-intercept of an exponential equation, you substitute 0 for the independent variable and solve for y. You could also find this by looking for a in the equation y=a(b^X).
You find the growth factor of an exponential relationship from a table by calculating what the dependent variable is multiplied by every time the independent variable increases by 1. To find the growth factor from a graph, find what the dependent variable is multiplied by every time the independent variable increases by 1. This is just like a table, only you use points on the line. The growth factor in the equation is b in the equation y=a(b^X).
1.b. The y-intercept is a in the equation y=a(b^X). The growth factor is b in this equation. To create an exponential relationship's equation, you simply substitute these numbers in for a and b in the equation.
2.a. In the equation y=a(b^X), a is the initial value of the exponential relationship, and b is the growth factor of the exponential relationship.
2.b. In a graph of y=a(b^X), a is the y-intercept.
2.c. In a graph of y=a(b^X), b is the amount y is increased by every x, or the steepness of the line.
You find the growth factor of an exponential relationship from a table by calculating what the dependent variable is multiplied by every time the independent variable increases by 1. To find the growth factor from a graph, find what the dependent variable is multiplied by every time the independent variable increases by 1. This is just like a table, only you use points on the line. The growth factor in the equation is b in the equation y=a(b^X).
1.b. The y-intercept is a in the equation y=a(b^X). The growth factor is b in this equation. To create an exponential relationship's equation, you simply substitute these numbers in for a and b in the equation.
2.a. In the equation y=a(b^X), a is the initial value of the exponential relationship, and b is the growth factor of the exponential relationship.
2.b. In a graph of y=a(b^X), a is the y-intercept.
2.c. In a graph of y=a(b^X), b is the amount y is increased by every x, or the steepness of the line.
Mathematical Reflection 2
1. a. You can use a table to find the y-intercept and growth factor from an exponential relationship. To find the y-intercept you look or extend the table to when x is 0. You can find the growth factor by finding the relationship between the y section in the table.
b. You use the y-intercept and growth factor to form an equation. You use the form, y=a(b^x). A is the y-intercept and b is the growth factor. You plug the numbers in and form an equation.
2. a. In an exponential relationship you use the equation y=a(b^x). The A is the y-intercept (initial value). B is the growth factor.
b. A is represented because it is the y-intercept .
c. B is represented because it is the growth factor, so it affects the steepness of the curve. It is sort-of like the slope of a line.
b. You use the y-intercept and growth factor to form an equation. You use the form, y=a(b^x). A is the y-intercept and b is the growth factor. You plug the numbers in and form an equation.
2. a. In an exponential relationship you use the equation y=a(b^x). The A is the y-intercept (initial value). B is the growth factor.
b. A is represented because it is the y-intercept .
c. B is represented because it is the growth factor, so it affects the steepness of the curve. It is sort-of like the slope of a line.
Monday 3/22/10: Fractional Growth Factors
Today in class, we went over the Exploring Exponential Functions worksheet we started on Friday. We determined the effect of a, b, and c of the graph of the equation y=a(b^x)+c:
We were introduced to fractional growth factors today. We saw that whether the growth factor was fractional or not did not effect the form of the equation. We did a problem in class about rabbits. We were given a table of the growth of a particular rabbit population. Using the information, we found that the growth rate was about 1.8. This did not influence the appication of y=a(b^x) to its graph.
- a is the starting point of the equation. It is the original y-intercept. When a is a negative number,the line of the equation points downward. On the other hand, when a is a positive number, the resulting line points up, giving it a shape similar to an U.
- b is the growth rate of the equation. It defines the steepness of the graph.
- c shifts the curved line up or down on the y-axis. When added to the a value of the equation, it results in the final y-intercept of the graph.
We were introduced to fractional growth factors today. We saw that whether the growth factor was fractional or not did not effect the form of the equation. We did a problem in class about rabbits. We were given a table of the growth of a particular rabbit population. Using the information, we found that the growth rate was about 1.8. This did not influence the appication of y=a(b^x) to its graph.
Mathematical Reflections 2
1.
a. The growth factor in an exponential line is how steep the curve is. The y-intercept is where the line starts on the y-axis. In a table, you divide the y of any x by the y of the x before it, and you get the growth factor. In a graph, you do the same, only you have to find the points, and divide the y values. In the equation y= a(b^x), b is the growth factor. To find the y-intercept in a table, you just look to where the x=o. On a graph, you just look at where the line crosses the y-axis. In the equation y=a(b^x), the y-intercept is a.
b. If you know what the y-intercept and growth factor are, you can plug them into the standard equation y= a(b^x).
2.
a. The a is the y-intercept, the b is the growth factor.
b. a is the y when x=o, and where the line crosses the y-axis
c. b is the steepness of the line.
a. The growth factor in an exponential line is how steep the curve is. The y-intercept is where the line starts on the y-axis. In a table, you divide the y of any x by the y of the x before it, and you get the growth factor. In a graph, you do the same, only you have to find the points, and divide the y values. In the equation y= a(b^x), b is the growth factor. To find the y-intercept in a table, you just look to where the x=o. On a graph, you just look at where the line crosses the y-axis. In the equation y=a(b^x), the y-intercept is a.
b. If you know what the y-intercept and growth factor are, you can plug them into the standard equation y= a(b^x).
2.
a. The a is the y-intercept, the b is the growth factor.
b. a is the y when x=o, and where the line crosses the y-axis
c. b is the steepness of the line.
Yesterday, our class worked on graphing Exponential Functions in order to learn how certain variables effect the function's graph. In the equation y=a(b)*+c the variables a, b and c all change the way the graph looks. The a is the y-intercept or initial value, unless there is a c in the equation. If there is a c, the two variables values are added to create the y-intercept. The b in the equation is the growth factor. It represents the rate that the line is increasing or decreasing much like the slope does in a linear equation. For example, if the b is two, then the y is doubling every interval. Next, the c is, again, the added or subtracted variable that is able to change the y-intercept and the a. The c sometimes is the y-intercept when you add one to it, because in many Exponential Functions the initial factor (a) is one. These three important variable can reveal a lot of information about an Exponential Function's graph.
Mathematical reflections 2
1. a. You can find the y-intercept in an exponential relationship by using a table, a graph, and an equation. You can find the y-intercept from a table by looking at the y when the x is zero on the table. Whatever the y is will be the y-intercept. you can find the y-intercept from a graph by looking at where the line passes the y axis and that will be the y-intercept. You can find the y-intrcept from an equation by looking at the number that is being multiplied by a number with an exponent and that will be the y-intercept.
You can also find the growth factor in an exponential relationship by using a table, a graph, and an equation. you can find the growth factor from a table by looking at ho much y is being multiplied by for each x and that will be the growth factor. You can find the growth factor from a graph by looking at how much y is multiplied on the graph for every x and that will be the growth factor. You can find the growth factor of an equation by seeing what number has an exponent and that will be the growth factor.
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship. You can put the y-intercept first and have it multiplied by the growth factor with an exponent of whatever number x the problem is at.
2. a. In the equation y=a(b^x), a is the initial value of the exponetial relationship and b is the growth factor of the exponetial relationship.
2. b. In a graph of y=a(b^x), a is the y-intercept.
2. c. In a graph of y=a(b^x), b is the amount y goes up for every time x goes up one interval or the steepness of the line.
You can also find the growth factor in an exponential relationship by using a table, a graph, and an equation. you can find the growth factor from a table by looking at ho much y is being multiplied by for each x and that will be the growth factor. You can find the growth factor from a graph by looking at how much y is multiplied on the graph for every x and that will be the growth factor. You can find the growth factor of an equation by seeing what number has an exponent and that will be the growth factor.
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship. You can put the y-intercept first and have it multiplied by the growth factor with an exponent of whatever number x the problem is at.
2. a. In the equation y=a(b^x), a is the initial value of the exponetial relationship and b is the growth factor of the exponetial relationship.
2. b. In a graph of y=a(b^x), a is the y-intercept.
2. c. In a graph of y=a(b^x), b is the amount y goes up for every time x goes up one interval or the steepness of the line.
Math Reflection 2
1a. The growth factor of an expenentionally grown line is the steepness. It shows how fast it goes up for example doubling or tripleing. To find it in in a table you divide y when x equals 2 by y when x equals 1. To get it from a graph you do the same thing from the table exept you have to find the points. From the equation: Y= a(b to the x) the growth factor is the b. To find the y intercept in a table or graph you take the number when x is zero. To find it from the equation you fill in 0for x or take the a value.
1b. You fill in the Y intercept as the a value and the growth factor for the b value.
2a. A is the y intercept b is the growth factor.
2b. A is the spot where x equals 0.
2c. B is how rapidly the line is going up.
1b. You fill in the Y intercept as the a value and the growth factor for the b value.
2a. A is the y intercept b is the growth factor.
2b. A is the spot where x equals 0.
2c. B is how rapidly the line is going up.
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