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## Comparing Graphs of Parabolas

Okay, now what if you are asked to compare the graph of y = x2 with the graph of y = (x – 1)2? Let's compare what happens when you plug numbers (x-values) into y = (x – 1)2 with what happens when you plug numbers (x-values) into y = x2:

 y = x2 y = (x – 1)2 If x = 1, y = 1. If x = 1, y = 0. If x = 2, y = 4. If x = 2, y = 1. If x = 3, y = 9. If x = 3, y = 4. If x = 4, y = 16. If x = 4, y = 9.

The two equations have the same y-values, but they match up with different x-values because y = (x – 1)2 subtracts 1 before squaring the x-value. As a result, the graph of y = (x – 1)2 looks identical to the graph of y = x2 except that the base is shifted to the right (on the x-axis) by 1:

How would the graph of y = x2 compare with the graph of y = x2 – 1?

In order to find a y-value with y = x2, you square the x-value. In order to find a y-value with y = x2 – 1, square the x-value and then subtract 1. This means the graph of y = x2 – 1 looks identical to the graph of y = x2 except that the base is shifted down (on the y-axis) by 1:

## Word Problems Aren't a Problem

You can easily solve the word problems using quadratic equations. Let's look carefully at an example.

#### Example

You have a patio that is 8 ft. by 10 ft. You want to increase the size of the patio to 168 square ft. by adding the same length to both sides of the patio. Let x = the length you will add to each side of the patio. You find the area of a rectangle by multiplying the length times the width. The new area of the patio will be 168 square ft.

 (x + 8)(x + 10) = 168 FOIL the factors (x + 8)(x + 10). x2 + 10x + 8x + 80 = 168 Simplify. x2 + 18x + 80 = 168 Subtract 168 from both sides of the equation. x2 + 18x + 80 – 168 = 168 – 168 Simplify both sides of the equation. x2 + 18x – 88 = 0 Factor. (x + 22)(x – 4) = 0 Set each factor equal to zero. x + 22 = 0 and x – 4 = 0 Solve the equation. x + 22 = 0 Subtract 22 from both sides of the equation. x + 22 – 22 = 0 – 22 Simplify both sides of the equation. x = –22 Solve the equation. x – 4 = 0 Add 4 to both sides of the equation. x – 4 + 4 = 0 + 4 Simplify both sides of the equation. x = 4

Because this is a quadratic equation, you can expect two answers. The answers are 4 and –22. However, –22 is not a reasonable answer. You cannot have a negative length. Therefore, the only answer is 4.

To check your calculations, review the original dimensions of the patio—8 ft. by 10 ft. If you were to add 4 to each side, the new dimensions would be 12 ft. by 14 ft. When you multiply 12 times 14, you get 168 square ft., which is the new area you wanted.

## What is the Quadratic Formula?

You can use the quadratic formula to solve quadratic equations. You might be asking yourself, "Why do I need to learn another method for solving quadratic equations when I already know how to solve them by using factoring?" Well, not all quadratic equations can be solved using factoring. You use the factoring method because it is faster and easier, but it will not always work. However, the quadratic formula will always work.

The quadratic formula is a formula that allows you to solve any quadratic equation—no matter how simple or difficult. If the equation is written in the form ax2 + bx + c = 0, then the two solutions for x will be x = . It is the ± in the formula that gives us the two answers: one with + in that spot, and one with –. The formula contains a radical, which is one of the reasons you studied radicals in the previous lesson. To use the formula, you substitute the values of a, b, and c into the formula and then carry out the calculations.

#### Example

 3x2 – x – 2 = 0 Determine a b, and c. a = 3, b = –1, and c = –2 Take the quadratic formula. Substitute in the values of a, b, and c. Simplify. Simplify more. Take the square root of 25. The solutions are 1 and

Tip:  To use the quadratic formula, you need to know the a, b, and c of the equation. However, before you can determine what a, b, and c are, the equation must be in ax2+ bx + c = 0 form. For example, the equation 5x2 + 2x = 9 must be transformed to ax2+ bx + c = 0 form.

Some equations will have radicals in their answers. The strategy for solving these equations is the same as the equations you just completed.

#### Example

 3m2 – 3m = 1 Subtract 1 from both sides of the equation. 3m2 – 3m – 1 = 1 – 1 Simplify. 3m2 – 3m – 1 = 0 Use the quadratic formula with a = 3, b = –3, and c = –1. Substitute the values for a, b, and c. Simplify. Simplify.

The solution to the equation is m = because one answer is m = and the other answer is m = .

Find practice problems and solutions for these concepts at Quadratic Trinomials, Quadratic Equations, and the Quadratic Formula Practice Problems.

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