Answers (Lesson 10-1)

Glencoe Algebra 1

A2

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NAME ______________________________________________ DATE ____________ PERIOD _____

10-1 Study Guide and Intervention

Graphing Quadratic Functions

Graph Quadratic Functions

Quadratic Function

a function described by an equation of the form f (x ) ax 2 bx c, where a 0

Example: y 2x 2 3x 8

The degree of a quadratic function is 2, and the exponents are positive. Graphs of quadratic functions have a general shape called a parabola. A parabola opens upward and has a minimum point when the value of a is positive, and a parabola opens downward and has a maximum point when the value of a is negative.

Example 1 Use a table of values to

graph y x2 4x 1.

xy 1 6 01 1 2 2 3 3 2 41

y

O

x

Graph the ordered pairs in the table and connect them with a smooth curve.

Example 2 Use a table of values to

graph y x2 6x 7.

xy 6 7 5 2 4 1 3 2 2 1 1 2 0 7

y

O

x

Graph the ordered pairs in the table and connect them with a smooth curve.

Exercises

Use a table of values to graph each function.

1. y x2 2

y

2. y x2 4

y

O

x

3. y x2 3x 2

y

O

x

O

x

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Glencoe Algebra 1

Lesson 10-1

NAME ______________________________________________ DATE ____________ PERIOD _____

10-1 Study Guide and Intervention (continued)

Graphing Quadratic Functions

Symmetry and Vertices Parabolas have a geometric property called symmetry. That

is, if the figure is folded in half, each half will match the other half exactly. The vertical line

containing the fold line is called the axis of symmetry.

Axis of

For the parabola y ax 2 bx c, where a 0, Example: The axis of symmetry of

Symmetry the line x 2ba is the axis of symmetry.

y x 2 2x 5 is the line x 1.

The axis of symmetry contains the minimum or maximum point of the parabola, the vertex.

Example Consider the graph of y 2x2 4x 1.

a. Write the equation of the axis of symmetry.

In y 2x2 4x 1, a 2 and b 4. Substitute these values into the equation of the axis of symmetry.

x

b 2a

x

4 2(2)

1

The axis of symmetry is x 1.

b. Find the coordinates of the vertex.

Since the equation of the axis of symmetry is x 1 and the vertex lies on the axis, the x-coordinate of the vertex is 1.

y 2x2 4x 1 y 2(1)2 4(1) 1 y 2(1) 4 1 y 1

Original equation Substitute. Simplify.

The vertex is at (1, 1).

c. Identify the vertex as a maximum or a minimum.

Since the coefficient of the x2-term is positive, the parabola opens upward, and the vertex is a minimum point.

d. Graph the function.

y

x ?1

O

x

(?1, ?1)

Exercises

Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or a minimum. Then graph the function.

1. y x2 3

x 0; (0, 3); min

y

2. y x2 4x 4

x 2; (2, 0); max

y

3. y x2 2x 3

x 1; (1, 2); min

y

O

x

O

x

? Glencoe/McGraw-Hill

580

O

x

Glencoe Algebra 1

Answers (Lesson 10-1)

? Glencoe/McGraw-Hill

NAME ______________________________________________ DATE ____________ PERIOD _____

10-1 Skills Practice

Graphing Quadratic Functions

Use a table of values to graph each function.

1. y x2 4

y

2. y x2 3

y

O

x

O

x

NAME ______________________________________________ DATE ____________ PERIOD _____

10-1 Practice (Average)

Graphing Quadratic Functions

Use a table of values to graph each function.

1. y x2 2

2. y x2 6x 3

y

y

3. y 2x2 8x 5

y

O

x

O

x

Ox

Lesson 10-1

Answers (Lesson 10-1)

A3

3. y x2 2x 6

y

4. y x2 4x 1

O

x

y

O

x

Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function.

5. y 2x2

x 0; (0, 0); min

y

6. y x2 2x 5

x 1; (1, 6); min

y

7. y x2 4x 1

x 2; (2, 3); max

y

O

x

O

x

O

x

8. y x2 2x 2

x 1; (1, 3); max

y

9. y 2x2 4x 2

10. y 2x2 4x 6

x 1; (1, 4); min x 1; (1, 8); max

y

y

O

x

O

x

? Glencoe/McGraw-Hill

581

O

x

Glencoe Algebra 1

Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function.

4. y x2 3

x 0; (0, 3); max

5. y 2x2 8x 3

x 2; (2, 5); max

6. y 2x2 8x 1

x 2; (2, 7); min

y

y

y

O

x

O

x

O

x

PHYSICS For Exercises 7?9, use the following information. Miranda throws a set of keys up to her brother, who is standing on a third-story balcony with his hands 38 feet above the ground. If Miranda throws the keys with an initial velocity of 40 feet per second, the equation h 16t2 40t 5 gives the height h of the keys after t seconds.

7. How long does it take the keys to reach their highest point? 1.25 s

8. How high do the keys reach? 30 ft

9. Will her brother be able to catch the keys? Explain. No, the keys will be 8 ft short of their target.

BASEBALL For Exercises 10?12, use the following information. A player hits a baseball at a 45? angle with the ground with an initial velocity of 80 feet per second from a height of three feet above the ground. The equation h 0.005x2 x 3 gives the path of the ball, where h is the height and x is the horizontal distance the ball travels.

10. What is the equation of the axis of symmetry? x 100

11. What is the maximum height reached by the baseball? 53 ft

12. An outfielder catches the ball three feet above the ground. How far has the ball traveled

horizontally when the outfielder catches it? 200 ft

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Glencoe Algebra 1

Answers

Glencoe Algebra 1

Glencoe Algebra 1

A4

? Glencoe/McGraw-Hill

NAME ______________________________________________ DATE ____________ PERIOD _____

10-1 Reading to Learn Mathematics

Graphing Quadratic Functions

Pre-Activity How can you coordinate a fireworks display with recorded music?

Read the introduction to Lesson 10-1 at the top of page 524 in your textbook. According to the graph, at what height does the rocket explode and in how

many seconds after being launched? It explodes at a height of 80 meters 4 seconds after being launched.

Reading the Lesson 1. The standard form for a quadratic function is y ax2 bx c. For the

function y 2x2 5x 3, the value of a is 2 , the value of b is 5 , and the value of c is 3 .

2. The graphs of two quadratic functions are shown below. Complete each statement about the graphs.

A.

y

B.

y

O

x

O

x

a. Each graph is a curve called a parabola .

b. The highest point of graph A is located at

(1, 4) . This point is the

maximum (maximum/minimum) point of the graph.

c. The lowest point of graph B is located at (1, 2) . This point is the

minimum (maximum/minimum) point of the graph.

3. The maximum or minimum point of a parabola is called the

vertex

of

the parabola.

4. If you fold a parabola along a line to get two halves that match exactly, the line where

you fold the parabola is the axis of symmetry of the parabola. This line goes

through the

vertex

of the parabola.

5. For a quadratic function y ax2 bx c, the parabola opens upward if a 0. It opens downward if a 0.

Helping You Remember

6. Look up the word vertex in a dictionary. You will find that it comes from the Latin word vertere, which means to turn. How can you use the idea of "to turn" to remember what

the vertex of a parabola is? Sample answer: The vertex of a parabola is the point at which the parabola turns upward or downward.

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Glencoe Algebra 1

Lesson 10-1

NAME ______________________________________________ DATE ____________ PERIOD _____

10-1 Enrichment

Translating Quadratic Graphs

When a figure is moved to a new position without undergoing any rotation, then the figure is said to have been translated to that position.

The graph of a quadratic equation in the form y (x b)2 c is a translation of the graph of y x2.

Start with y x2.

Slide to the right 4 units. y (x 4)2

Then slide up 3 units. y (x 4)2 3

y

O

x

These equations have the form y x2 c. Graph each equation.

1. y x2 1

y

2. y x2 2

y

3. y x2 2

y

O

x

O

x

O

x

These equations have the form y (x b)2. Graph each equation.

4. y (x 1)2

y

5. y (x 3)2

y

6. y (x 2)2

y

O

x

? Glencoe/McGraw-Hill

O

x

584

O

x

Glencoe Algebra 1

Answers (Lesson 10-1)

Glencoe Algebra 1

A5

? Glencoe/McGraw-Hill

NAME ______________________________________________ DATE ____________ PERIOD _____

10-2 Study Guide and Intervention

Solving Quadratic Equations by Graphing

Solve by Graphing

Quadratic Equation an equation of the form ax 2 bx c 0, where a 0

The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by graphing the related quadratic function f(x) ax2 bx c and finding the x-intercepts or zeros of the function.

Example 1 Solve x2 4x 3 0 by

graphing.

Graph the related function f(x) x2 4x 3.

The equation of the axis of symmetry is

x

4 2(1)

or

2.

The

vertex

is

at

(

2,

1).

Graph the vertex and several other points on

either side of the axis of symmetry.

f (x)

Example 2 Solve x2 6x 9 0 by

graphing.

Graph the related function f(x) x2 6x 9.

The equation of the axis of symmetry is

x

6 2(1)

or

3.

The

vertex

is

at

(

3,

0).

Graph

the vertex and several other points on either

side of the axis of symmetry.

f (x)

O

x

O

x

To solve x2 4x 3 0, you need to know where the value of f(x) 0. This occurs at the x-intercepts, 3 and 1.

The solutions are 3 and 1.

To solve x2 6x 9 0, you need to know where the value of f(x) 0. The vertex of the parabola is the x-intercept. Thus, the only solution is 3.

Exercises

Solve each equation by graphing.

1. x2 7x 12 0

f (x)

2. x2 x 12 0

4 f(x)

O

x

8 4 O 4 8

12

4 8x

3; 4

4, 3

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585

3. x2 4x 5 0

f (x)

O

x

no real roots

Glencoe Algebra 1

Answers

Lesson 10-2

NAME ______________________________________________ DATE ____________ PERIOD _____

10-2 Study Guide and Intervention (continued)

Solving Quadratic Equations by Graphing

Estimate Solutions The roots of a quadratic equation may not be integers. If exact

roots cannot be found, they can be estimated by finding the consecutive integers between which the roots lie.

Example Solve x2 6x 6 0 by graphing. If integral roots cannot be found,

estimate the roots by stating the consecutive integers between which the roots lie.

Graph the related function f(x) x2 6x 6.

f (x)

x f (x) Notice that the value of the function changes from negative to positive between the x-values

5 1 of 5 and 4 and between 2 and 1.

4 2

O

x

3 3

2 2

1 1

The x-intercepts of the graph are between 5 and 4 and between 2 and 1. So one root is between 5 and 4, and the other root is between 2 and 1.

Exercises

Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.

1. x2 7x 9 0

2. x2 x 4 0

3. x2 4x 6 0

f (x)

f (x)

f (x)

O

x

6 x 5, 2 x 1

4. x2 4x 1 0

f (x)

O

x

O

x

2 x 1, 2x3

5. 4x2 12x 3 0

f (x)

O

x

O

x

no real roots

6. x2 2x 4 0

f (x)

O

x

1 x 0, 4x5

? Glencoe/McGraw-Hill

0 x 1, 2x3

586

2 x 1, 3x4

Glencoe Algebra 1

Answers (Lesson 10-2)

Glencoe Algebra 1

A6

? Glencoe/McGraw-Hill

NAME ______________________________________________ DATE ____________ PERIOD _____

10-2 Skills Practice

Solving Quadratic Equations by Graphing

Solve each equation by graphing.

1. x2 4x 5 0

2. c2 6c 8 0 4, 2

f (x)

f (c)

O

x

3. a2 2a 1 1

f (a)

Oc

4. n2 7n 10 2, 5

f (n)

O

a

O

n

Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.

5. p2 4p 2 0

6. x2 x 3 0

f (p)

f (x)

O

p

4 p 3, 1 p 0

7. d2 6d 3

f (d )

O

d

6 d 5, 1 d 0

? Glencoe/McGraw-Hill

O

x

3 x 2, 1 x 2

8. h2 1 4h

f (h)

O

h

0 h 1, 3 h 4

587

Glencoe Algebra 1

Lesson 10-2

NAME ______________________________________________ DATE ____________ PERIOD _____

10-2 Practice (Average)

Solving Quadratic Equations by Graphing

Solve each equation by graphing.

1. x2 5x 6 0 2, 3

2. w2 6w 9 0 3

f (x)

f (w)

3. b2 4b 5 0

f (b)

O

x

O

w

O

b

Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.

4. p2 4p 3

5. 2m2 5 10m

6. 2v2 8v 7

f (p)

f (m)

f (v)

O

p

O

m

O

v

5 p 4, 0p1

0 m 1, 4m5

3 v 2, 2 v 1

NUMBER THEORY For Exercises 7 and 8, use the following information.

Two numbers have a sum of 2 and a product of 8. The quadratic equation n2 2n 8 0 can be used to determine the two numbers.

7. Graph the related function f (n) n2 2n 8 and

determine its x-intercepts. 2, 4

8. What are the two numbers? 2 and 4

f (n)

O

n

DESIGN For Exercises 9 and 10, use the following information.

A footbridge is suspended from a parabolic support. The function

h(x)

1 25

x2

9

represents

the

height

in

feet

of

the

support

above

the walkway, where x 0 represents the midpoint of the bridge.

9. Graph the function and determine its x-intercepts. 15, 15

10. What is the length of the walkway between the two supports?

30 ft

12 h(x) 6

12 6 O 6

12

6 12 x

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Glencoe Algebra 1

Answers (Lesson 10-2)

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