Chapter 3: Quadratic Functions and Equations; Inequalities. 35. 7i(2 â 5i). = 14i â 35i2. Using the distributive law. = 14i + 35 i2 = â1. = 35 + 14i. Writing in the form a + bi. 36. 3i(6 + 4i) = 18i + 12i2 = 18i â 12 = â12 + 18i. 37. â2i(

Polynomial functions are classified by degree. For instance, a constant func- tion has degree 0 and a linear function has degree 1. In this section, you will study second-degree polynomial functions, which are called quadratic functions. For instance

SAVINGS Jack has $100 to buy a game system. He plans ...... DEMOLITION DERBY When a car hits an object, the damage is measured by the collision impact.

square root, p. 266. â¢ complex number, p. 276. â¢ imaginary number, p. 276. â¢ completing the square, p. 284. â¢ quadratic formula, p. 292. â¢ discriminant, p. 294. â¢ best-fitting quadratic model ..... Drawing a Graph Use the table to graph t

MCR3UW. Equations of. Quadratic. Functions. Homework. Exercise. 2J. Use the information provided in the graphs to write the equation of each function in standard form y ax2. + bx. Ã· c. 1. 2 y. 4. 7. 3 yl. (2,. 5. (-1,0). 6. (6,0). X. (1,13). (4,-5).

Online Option Take an online self-check Chapter Readiness Quiz at connectED.mcgraw-hill.com. Diagnose ..... 2A. rn + 5n - r - 5. 2B. 3np + 15p - 4n - 20 ...... degree. Example 1 Factor a x2 + bx + c. Factor each trinomial. a. 7 x 2 + 29x + 4.

coefficient of the first term of a polynomial written in standard form ...... Original equation x 2 - 2x - 8 = 0. (-2) 2 - 2(-2) - 8 0 x = -2 or x = 4. (4) 2 - 2(4) - 8 0. 0 = 0. Simplify. 0 = 0. GuidedPractice. Solve each equation by graphing. 1A. -

zero feature to answer the questions. EXTRA PRACTICE for Lesson 4.1, p. 1013. ONLINE QUIZ at classzone.com. Solve the equation. (p. 18). 62. x 2 3 5 0 ...... systems of linear equations in three variables, see p. 178. Write a quadratic function in st

3.1. 312 Chapter 3 Polynomial and Rational Functions. Quadratic Functions. Many sports involve objects that are thrown, kicked, or hit, and then proceed with no addi- .... Solve for Add to both sides of the equation. Divide both sides by 2. Apply the

10

Quadratic Equations and Functions 10.1

Graph y 5 ax 2 1 c

10.2 Graph y 5 ax 2 1 bx 1 c 10.3 Solve Quadratic Equations by Graphing 10.4

Use Square Roots to Solve Quadratic Equations

10.5 Solve Quadratic Equations by Completing the Square 10.6

Solve Quadratic Equations by the Quadratic Formula

10.7

Interpret the Discriminant

10.8 Compare Linear, Exponential, and Quadratic Models

Before In previous chapters, you learned the following skills, which you’ll use in Chapter 10: reflecting points in a line and finding square roots.

Prerequisite Skills VOCABULARY CHECK Copy and complete the statement. 1. The x-coordinate of a point where a graph crosses the x-axis is a(n) ? . 2. A(n) ? is a function of the form y 5 a p bx where a Þ 0, b > 0, and b Þ 1.

SKILLS CHECK Draw the blue figure. Then draw its image after a reflection in the red line. (Review p. 922 for 10.1–10.3.)

3.

4.

y

1

5.

y

1

x 1

x 1

}

}

7. 2Ï 25

}

8. Ï 1

1SFSFRVJTJUFTLJMMTQSBDUJDFBUDMBTT[POFDPN

626

1

x 1

Evaluate the expression. (Review p. 110 for 10.4–10.6.) 6. Ï 81

y

}

9. 6Ï 64

Now In Chapter 10, you will apply the big ideas listed below and reviewed in the Chapter Summary on page 695. You will also use the key vocabulary listed below.

Big Ideas 1 Graphing quadratic functions 2 Solving quadratic equations 3 Comparing linear, exponential, and quadratic models KEY VOCABULARY • quadratic function, p. 628

• vertex, p. 628

• quadratic equation, p. 643

• parabola, p. 628

• axis of symmetry, p. 628

• completing the square, p. 663

• parent quadratic function, p. 628

• minimum value, p. 636

• quadratic formula, p. 671

• maximum value, p. 636

• discriminant, p. 678

Why? You can use a quadratic model for real-world situations involving vertical motion. For example, you can write and solve a quadratic equation to find the time a snowboarder is in the air during a jump.

Algebra The animation illustrated below for Exercise 50 on page 668 helps you answer this question: How many seconds is the snowboarder in the air during a jump?

You need to find the time that the snowboarder is in the air.

#HECK!NSWER

Click the buttons and enter expressions to solve the equation.

Algebra at classzone.com Other animations for Chapter 10: pages 634, 636, 642, 662, 668, 672, 684, and 695

627

10.1 Before Now Why?

Key Vocabulary • quadratic function • parabola • parent quadratic function • vertex • axis of symmetry

Graph y 5 ax2 1 c You graphed linear and exponential functions. You will graph simple quadratic functions. So you can solve a problem involving an antenna, as in Ex. 40.

A quadratic function is a nonlinear function that can be written in the standard form y 5 ax2 1 bx 1 c where a Þ 0. Every quadratic function has a U-shaped graph called a parabola. In this lesson, you will graph quadratic functions where b 5 0.

For Your Notebook

KEY CONCEPT Parent Quadratic Function

The most basic quadratic function in the family of quadratic functions, called the parent quadratic function, is y 5 x2. The graph of y 5 x2 is shown below. The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry. The axis of symmetry for the graph of y 5 x2 is the y-axis, x 5 0.

y

The lowest or highest point on a parabola is the vertex. The vertex of the graph of y 5 x2 is (0, 0).

EXAMPLE 1

y 5 x2 1 1

(0, 0)

x

Graph y 5 ax 2 where ⏐a⏐ > 1

STEP 1 Make a table of values for y 5 3x2.

y

x

22

21

0

1

2

y

12

3

0

3

12

PLOT ADDITIONAL POINTS

STEP 2 Plot the points from the table.

If you are having difficulty seeing the shape of the parabola, plot additional points.

STEP 4 Compare the graphs of y 5 3x2 and y 5 x2 .

628

STEP 3 Draw a smooth curve through the points. Both graphs open up and have the same vertex, (0, 0), and axis of symmetry, x 5 0. The graph of y 5 3x2 is narrower than the graph of y 5 x2 because the graph of y 5 3x2 is a vertical stretch (by a factor of 3) of the graph of y 5 x2.

Chapter 10 Quadratic Equations and Functions

y 5 3x2

4

y 5 x2 1

x

EXAMPLE 2

Graph y 5 ax 2 where ⏐a⏐ < 1

1 2 Graph y 5 2} x . Compare the graph with the graph of y 5 x 2 . 4

y

STEP 1 Make a table of values for y 5 2}1x2. 4

MAKE A TABLE To make the calculations easier, choose values of x that are multiples of 2.

x

24

22

0

2

4

y

24

21

0

21

24

y 5 x2

1

3

STEP 2 Plot the points from the table.

x

y 5 2 14 x 2

STEP 3 Draw a smooth curve through the points.

STEP 4 Compare the graphs of y 5 2}1x2 and y 5 x2 . Both graphs have the 4

same vertex (0, 0), and the same axis of symmetry, x 5 0. However, 1

the graph of y 5 2} x2 is wider than the graph of y 5 x2 and it opens 4 1

down. This is because the graph of y 5 2} x2 is a vertical shrink 4

1 by a factor of }14 2 with a reﬂection in the x-axis of the graph of y 5 x . 2

GRAPHING QUADRATIC FUNCTIONS Examples 1 and 2 suggest the following general result: a parabola opens up when the coefficient of x2 is positive and opens down when the coefficient of x2 is negative.

EXAMPLE 3

Graph y 5 x 2 1 c

Graph y 5 x 2 1 5. Compare the graph with the graph of y 5 x 2 .

STEP 1 Make a table of values for y 5 x2 1 5. x

22

21

0

1

2

y

9

6

5

6

9

y

STEP 2 Plot the points from the table.

y 5 x2 1 5

STEP 3 Draw a smooth curve through the points. STEP 4 Compare the graphs of y 5 x2 1 5 and y 5 x2 . Both graphs open up and have the same axis of symmetry, x 5 0. However, the vertex of the graph of y 5 x2 1 5, (0, 5), is different than the vertex of the graph of y 5 x2, (0, 0), because the graph of y 5 x2 1 5 is a vertical translation (of 5 units up) of the graph of y 5 x2.

✓

GUIDED PRACTICE

2

y 5 x2 1

x

for Examples 1, 2, and 3

Graph the function. Compare the graph with the graph of y 5 x 2 . 1. y 5 24x2

1 2 2. y 5 } x 3

3. y 5 x2 1 2

10.1 Graph y 5 ax 2 1 c

629

Graph y 5 ax 2 1 c

EXAMPLE 4

1 2 Graph y 5 } x 2 4. Compare the graph with the graph of y 5 x 2 . 2

y

STEP 1 Make a table of values for y 5 }1x2 2 4. 2

y 5 x2

x

24

22

0

2

4

y

4

22

24

22

4

1 1

x

STEP 2 Plot the points from the table. y 5 12 x 2 2 4

STEP 3 Draw a smooth curve through the points.

STEP 4 Compare the graphs of y 5 }1x2 2 4 and y 5 x2 . Both graphs open up 2

and have the same axis of symmetry, x 5 0. However, the graph of 1 2 y5} x 2 4 is wider and has a lower vertex than the graph of y 5 x2 2

1 2 because the graph of y 5 } x 2 4 is a vertical shrink and a vertical 2

translation of the graph of y 5 x2.

✓

GUIDED PRACTICE

for Example 4

Graph the function. Compare the graph with the graph of y 5 x 2 . 4. y 5 3x2 2 6