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.
Subscribe to:
Posts (Atom)
