Intercepts and Factoring - Weebly

[Pages:11]Intercepts and Factoring

To solve the unit problem, you had to find the maximum value for a quadratic function. To do so, you learned about the vertex form of quadratic expressions. You also used vertex form to find x-intercepts for quadratic graphs. These are points where the graph meets the x-axis. Now you will look at another way to find x-intercepts as well as points where the graph intersects other horizontal lines. Finding these points involves solving quadratic equations. The vertex form will allow you to solve any quadratic equation that has solutions--including those with answers that require square roots. But using vertex form isn't always the simplest approach. In the final activities of this unit, you will see that factoring is convenient for solving some quadratic equations.

Alida Jekabson finds a point of intersection.

46 INTERACTIVE MATHEMATICS PROGRAM

Factoring

Group Activity

The quadratic function y (x 2)(x 3) is written in factored form as a product of two linear expressions. Factored form is useful for finding x-intercepts. The x-intercepts are the values of x that make y 0.

What values of x will make the product (x 2)(x 3) equal to zero? If 2 is substituted for x, you get (2 2)(2 3) 0(5) 0. If 3 is substituted for x, you get (3 2)(3 3) 5(0) 0.

The function y (x 2)(x 3) in standard form is y x2 x 6. So the x-intercepts for the function y x2 x 6 are at (2, 0) and (3, 0).

To find the x-intercepts of a quadratic in standard form, y ax2 bx c, you can rewrite the function in factored form.

Rewrite each quadratic function in factored form. An area model, like the one shown in Question 1, may be helpful.

1. y x2 6x 8

x

?

2. y x2 9x 18 3. y x2 11x 18 4. y x2 10x 24

x x2

?x

? ?x

8

5. y x2 10x 24

Fireworks: Intercepts and Factoring 47

Let's Factor!

Activity

1. Try to write each quadratic expression as a product of two linear expressions. That is, try to write each expression in factored form.

a. x2 5x 6 b. x2 2x 15

c. x2 6x 10 d. x2 9x 8

e. x2 16

f. x2 10x 6

2. Find the x-intercepts, if any, for each quadratic function. For functions that have x-intercepts, sketch a graph using the intercepts and one or two other points

that satisfy the equation.

a. y x2 5x 6 b. y x2 2x 15

c. y x2 6x 10 d. y x2 9x 8

e. y x2 16

f. y x2 10x 6

3. Find the x-intercepts of each function. Sketch a graph of the function using the x-intercepts and one or two other points that satisfy the equation.

a. y (x 3)(x 3)(x 1) b. y (x 3)(x 3)(x 1)

c. y (x 2)(x2 9)

d. y x(x 2)(x 2)

48 INTERACTIVE MATHEMATICS PROGRAM

Solve That Quadratic!

Activity

Finding the x-intercepts of a quadratic function y x2 bx c is basically the same as solving the quadratic equation x2 bx c 0.

For example, the function y x2 5x 6 has x-intercepts at (6, 0) and (1, 0). So the solutions to the equation x2 5x 6 0 are the values 6 and 1.

1. Find the solutions to each quadratic equation. a. x2 7x 6 0 b. x2 3x 10 0 c. 2x2 8x 6 0 d. x2 4x 6 0

2. Dairyman Johnson has decided having a pen with the maximum area might not be best. After all, the bigger the pen, the more work it will be to clean it. He now thinks that 20,000 square feet is the best possible area for a cattle pen. He is still using a total of 500 feet of fencing, in addition to the fence along the border. a. Write an equation whose solution will give the distance the pen should extend out from the existing fence. b. Solve your equation and explain the results.

Border fence

Cattle pen

Total of 500 feet of fencing

Fireworks: Intercepts and Factoring 49

Quadratic Choices

Activity

You have worked with two main methods for finding the x-intercepts for quadratic functions.

? Using vertex form ? Using factored form

For each function, find the x-intercepts using the method you prefer. Explain why you chose that method.

1. y (x 3)2 25

2. y (x 9)(x 6)

3. y x2 12x 20

4. The choice of method is hardest when the function is given in standard form, y ax2 bx c. Explain what characteristics of the coefficients a, b, and c might make you choose one method over the other.

50 INTERACTIVE MATHEMATICS PROGRAM

A Quadratic Summary

Activity

Summarize what you've learned about quadratic expressions, quadratic functions, and quadratic equations. Include ideas about the graphs of quadratic functions.

In your summary, define the terms you use. Here are some ideas. Illustrate your ideas with specific examples. Also give examples of the algebraic techniques you have learned.

Quadratic functions Standard form Vertex form Factored form Parabola Vertex Intercepts

Quadratic equations

Fireworks: Intercepts and Factoring 51

Fireworks Portfolio

Activity

Now you will put together your portfolio for Fireworks. This process has three steps.

? Write a cover letter that summarizes the unit. ? Choose papers to include from your work in the unit. ? Discuss your personal growth during the unit.

Cover Letter

Look back over Fireworks. Describe the unit's central problem and main mathematical ideas. Your description should give an overview of how the key ideas were developed. You should also tell how these ideas were used to solve the central problem. As part of compiling your portfolio, you will select some activities that you think were important in developing the unit's key ideas. Your cover letter should explain why you selected each item.

52 INTERACTIVE MATHEMATICS PROGRAM

continued L

Selecting Papers

Your portfolio for Fireworks should contain these items. ? One or two activities that helped you understand the value of vertex

form in solving real-world problems ? One or two activities that helped you become comfortable with the

mechanics of working with quadratic expressions ? A Fireworks Summary ? A Quadratic Summary ? One of the two POWs you completed during this unit: Growth of Rat

Populations or Twin Primes

What About All That Algebra?

This unit included considerable work with the mechanics of algebra. You transformed many algebraic expressions from one form into another. As part of your portfolio, write about your comfort level with these mechanics. Also discuss the degree to which you think these mechanics are useful or meaningful to you.

Fireworks: Intercepts and Factoring 53

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