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.
Sunday, May 9, 2010
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).
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