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.