Appendices © 2008 Pearson Addison-Wesley. All rights reserved.

Appendices

© 2008 Pearson Addison-Wesley.All rights reserved

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-2

Appendices

Appendix AEquations and Inequalities

Appendix BGraphs of Equations

Appendix CFunctions

Appendix DGraphing Techniques


Equations and InequalitiesAEquations ▪ Solving Linear Equations ▪ Solving Quadratic Equations ▪ Inequalities ▪ Solving Linear Inequalities ▪ Interval Notation ▪ Three-Part Inequalities


A Example 1 Solving a Linear Equation (page 423)

Solve .

Solution set: {6}

Distributive property

Combine terms.

Add 4 to both sides.Add 12x to both sides.Combine terms.Divide both sides by 4.


Solve .

A Example 2 Clearing Fractions Before Solving a Linear Equation (page 424)

Solution set: {–10}

Multiply by 10, the LCD of all the fractions.


Combine terms.

Add –4s and –6 to both sides. Combine terms.

Divide both sides by –6.


Decide whether the equation is an identity, a conditional equation, or a contradiction. Give the solution set.

A Example 3(a) Identifying Types of Equations (page 425)

This is a conditional equation.Solution set: {11}

Add –4x and 9 to both sides. Combine terms.

Divide both sides by 2.



This is a contradiction.Solution set: ø


Subtract 14x from both sides.

A Example 3(b) Identifying Types of Equations (page 425)



A Example 3(c) Identifying Types of Equations (page 425)

This is an identity.Solution set: {all real numbers}


Combine terms.

Add x and –3 to both sides.


A Example 4 Using the Zero-Factor Property (page 426)

Solve .

Factor.

Set each factor equal to 0 and then solve for x.

or

or

or


A Example 4 Using the Zero-Factor Property (cont.)

Now check.

Solution set:


A Example 5 Using the Square Root Property (page 426)

Solve each quadratic equation.

Generalized square root property

(a)

(b)


A Example 6 Using the Quadratic Formula (Real Solutions) (page 115)

Solve .

a = 1, b = 6, c = –3

Write the equation in standard form.

Quadratic formula


A Example 6 Using the Quadratic Formula (Real Solutions) (cont.)

Solution set:


Subtract 7.

Divide by –2. Reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

Solution set: {x|x > 6}

A Example 7 Solving a Linear Inequality (page 428)

Solve .


A Example 8 Solving a Three-Part Inequality (page 429)

Solve .

Add 8.

Divide by 6.

Solution set:


Graphs of EquationsBThe Rectangular Coordinate System ▪ The Pythagorean Theorem and the Distance Formula ▪ The Midpoint Formula ▪ Graphing Equations ▪ Circles


B Example 1 Using the Distance Formula (page 433)

Find the distance between P(3, –5) and Q(–2, 8).

The distance between P(3, –5) and Q(–2, 8) is


Find the coordinates of the midpoint M of the segment with endpoints (–7, –5) and (–2, 13).

B Example 2 Using the Midpoint Formula (page 434)

Midpoint formula:

The coordinates of the midpoint M of the segment

with endpoints (–7, –5) and (–2, 13) are


B Example 3(a) Finding Ordered Pairs That are Solutions to Equations (page 434)

x

–1

1

3

Three ordered pairs that are solutions are (−1, 7),

(1, 3), and (3, −1).

Find three ordered pairs that are solutions to y = –2x + 5.


Find three ordered pairs that are solutions to .

B Example 3(b) Finding Ordered Pairs That are Solutions to Equations (page 434)

x

–2

–1

2

Three ordered pairs that are solutions are (−2, −3),

(−1, 0), and (2, −3).


B Example 4(a) Graphing Equations (page 435)

Step 1: Find the x-intercept and the y-intercept.

Intercepts: and

Graph the equation y = –2x + 5. (See Classroom Example 3a.)


B Example 4(a) Graphing Equations (cont.)

Step 2: Use the other ordered pairs found in Example 3a: (−1, 7), (1, 3), and (3, −1).

Steps 3 and 4: Plot and then connect the five points.


Graph the equation . (See Classroom Example 3b.)

B Example 4(b) Graphing Equations (page 435)

Step 1: Find the x-intercept and the y-intercept.

Intercepts: (–1, 0), (1, 0) and (0, 1)


B Example 8(c) Graphing Equations (cont.)

Step 2: Use the other ordered pairs found in Example 3b: (−2, −3) and (2, −3).

Steps 3 and 4: Plot and then connect the five points.


Find the center-radius form of the equation of each circle described.

B Example 5 Find the Center-Radius Form (page 437)

(a) Center at (1, –2), radius 3

(b) Center at (0, 0), radius 2

(h, k) = (1, –2), r = 3

(h, k) = (0, 0), r = 2


B Example 6(a) Graphing Circles (page 437)

Graph

Write the equation as

(h, k) = (1, –2), r = 3


B Example 6(b) Graphing Circles (page 437)

Graph

(h, k) = (0, 0), r = 2

Write the equation as


FunctionsCRelations and Functions ▪ Domain and Range ▪ Determining Functions from Graphs or Equations ▪ Function Notation ▪ Increasing, Decreasing, and Constant Functions


C Example 1 Deciding Whether Relations Define Functions

(page 441)Decide whether the relation determines a function.

M is a function because each distinct x-value has exactly one y-value.

(a)


C Example 1 Deciding Whether Relations Define Functions

(cont.)

(b)

N is not a function because the x-value –4 has two y-values.


Give the domain and range of the relation. Is the relation a function?

{(–4, –2), (–1, 0), (1, 2), (3, 5)}

Domain: {–4, –1, 0, 3}

Range: {–2, 0, 2, 5}

The relation is a function because each x-value corresponds to exactly one y-value.

C Example 2(a) Finding Domains and Ranges of Relations

(page 442)


C Example 2(b) Finding Domains and Ranges of Relations

(cont.)

Give the domain and range of the relation. Is the relation a function?

Domain: {1, 2, 3}

Range: {4, 5, 6, 7}

The relation is not a function because the x-value 2 corresponds to two y-values, 5 and 6.


C Example 3(a) Finding Domains and Ranges from Graphs

(page 442)

Give the domain and range of the relation.

Domain: {–2, 4}

Range: {0, 3}


C Example 3(b) Finding Domains and Ranges from Graphs

(cont.)


Domain:

Range:



Domain: [–5, 5]

Range: [–3, 3]

C Example 3(c) Finding Domains and Ranges from Graphs

(cont.)


C Example 3(d) Finding Domains and Ranges from Graphs

(cont.)


Domain:

Range:


Determine if the relation is a function and give the domain and range.

y is found by multiplying x by 2 and subtracting 5.

C Example 4(a) Identifying Functions, Domains, and Ranges (page 444)

y = 2x – 5

Domain: Range:

Each value of x corresponds to just one value of y, so the relation is a function.



y is found by squaring x by 2 and adding 3.

C Example 4(b) Identifying Functions, Domains, and Ranges (cont.)

y = x2 + 3

Domain: Range:




For any choice of x in the domain, there are two possible values for y.

C Example 4(c) Identifying Functions, Domains, and Ranges (cont.)

x = |y|

Domain: Range:

The relation is not a function.



y is found by dividing 3 by x + 2.

C Example 4(d) Identifying Functions, Domains, and Ranges (cont.)

Domain: Range:



Let and .

Find f(–3), f(r), and g(r + 2).

C Example 5 Using Function Notation (page 445)


Find f(–1) for each function.

(a) f(x) = 2x2 – 9

(b) f = {(–4, 0), (–1, 6), (0, 8), (2, –2)}

C Example 6 Using Function Notation (page 446)

f(–1) = 2(–1)2 – 9 = –7

f(–1) = 6


Find f(–1) for each function.

(c) (d)

C Example 6 Using Function Notation (cont.)

f(–1) = 5 f(–1) = 0


The figure is the graph of a function. Determine the intervals over which the function is increasing, decreasing, or constant.

C Example 7 Determining Intervals Over Which a Function is Increasing, Decreasing, or Constant

(page 447)

Increasing on Decreasing on

Constant on


Graphing TechniquesDStretching and Shrinking ▪ Reflecting ▪ Symmetry ▪ Translations


D Example 1 Stretching or Shrinking a Graph (page 451)

x

–2

–1

0

1

2

4 1

1 ¼

0 0

1 ¼

4 1

Create a table of values.

Note that for corresponding values of x, the y-values of g(x) are each one-fourth that of f(x).

Graph the function


D Example 2 Reflecting a Graph Across an Axis (page 452)

x

–2

–1

0

1

2

2 –2

1 –1

0 0

1 –1

2 –2


Note that every y-value of g(x) is the negative of the corresponding y-value of f(x). The graph of f(x) is reflected across the x-axis to give the graph of g(x).

Graph the function


D Example 2 Reflecting a Graph Across an Axis (cont.)

x

–2

–1

0

1

2

2 2

1 1

0 0

1 1

2 2


Note that every y-value of g(x) is the same of the corresponding y-value of f(x). The graph of f(x) is reflected across the y-axis to give the graph of g(x).

Graph the function


D Example 3(a) Testing for Symmetry with Respect to an Axis (page 454)

Test for symmetry with respect to the x-axis and the y-axis.

Replace x with –x:

The result is not the same as the original equation.

The graph is not symmetric with respect to the y-axis.

Replace y with –y:

The result is the same as the original equation.

The graph is symmetric with respect to the x-axis.

The graph is symmetric with respect to the x-axis only.


D Example 3(a) Testing for Symmetry with Respect to an Axis (cont.)


D Example 3(b) Testing for Symmetry with Respect to an Axis (page 454)




The graph is symmetric with respect to the y-axis.


The result is the not same as the original equation.

The graph is not symmetric with respect to the x-axis.

The graph is symmetric with respect to the y-axis only.


D Example 3(b) Testing for Symmetry with Respect to an Axis (cont.)


D Example 3(c) Testing for Symmetry with Respect to an Axis (page 454)




The graph is not symmetric with respect to the y-axis.


The result is the not same as the original equation.

The graph is not symmetric with respect to the x-axis.

The graph is not symmetric with respect to either axis.


D Example 3(c) Testing for Symmetry with Respect to an Axis (cont.)


D Example 3(d) Testing for Symmetry with Respect to an Axis (page 454)




The graph is symmetric with respect to the y-axis.


The result is same as the original equation.

The graph is symmetric with respect to the x-axis.

The graph is symmetric with respect to both axes.


D Example 3(d) Testing for Symmetry with Respect to an Axis (cont.)


D Example 4(a) Testing for Symmetry with Respect to the Origin (page 455)

Is the graph of symmetric with respect to the origin?

Replace x with –x and y with –y:


The graph is symmetric with respect to the origin.


D Example 4(b) Testing for Symmetry with Respect to the Origin (page 455))

Is the graph of symmetric with respect to the origin?

Replace x with –x and y with –y:


The graph is not symmetric with respect to the origin.


D Example 5(a) Translating Graphs (page 455)

Graph .

Compare a table of values for with .

x

–2

–1

0

1

2

4 6

1 3

0 2

1 3

4 6

The graph of f(x) is the same as the graph of g(x) translated 2 units up.


D Example 5(b) Translating Graphs (page 455)

Graph .

x

–4

–3

–2

–1

0

1

16 4

9 1

4 0

1 1

0 4

Compare a table of values for with .

1 9


D Example 5(b) Translating Graphs (cont.)

Graph .

The graph of f(x) is the same as the graph of g(x) translated 2 units left.


D Example 6 Using More Than One Transformation on a Graph

(page 457)

Graph .

This is the graph of translated one unit to the right, reflected across the x-axis, and then translated four units up.

Appendices © 2008 Pearson Addison-Wesley. All rights reserved.

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