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.
Subscribe to:
Posts (Atom)
