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Appendices © 2008 Pearson Addison-Wesley. All rights reserved

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-2 Appendices Appendix AEquations and Inequalities Appendix BGraphs of Equations Appendix CFunctions Appendix DGraphing Techniques

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-3 Equations and Inequalities A Equations ▪ Solving Linear Equations ▪ Solving Quadratic Equations ▪ Inequalities ▪ Solving Linear Inequalities ▪ Interval Notation ▪ Three-Part Inequalities

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-4 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-5 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. Distributive property Combine terms. Add –4s and –6 to both sides. Combine terms. Divide both sides by –6.

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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-7 Decide whether the equation is an identity, a conditional equation, or a contradiction. Give the solution set. This is a contradiction. Solution set: ø Distributive property Subtract 14x from both sides. A Example 3(b) Identifying Types of Equations (page 425)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-8 Decide whether the equation is an identity, a conditional equation, or a contradiction. Give the solution set. A Example 3(c) Identifying Types of Equations (page 425) This is an identity. Solution set: {all real numbers} Distributive property Combine terms. Add x and –3 to both sides.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-9 A Example 4 Using the Zero-Factor Property (page 426) Solve. Factor. Set each factor equal to 0 and then solve for x. or

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-10 A Example 4 Using the Zero-Factor Property (cont.) Now check. Solution set:

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-11 A Example 5 Using the Square Root Property (page 426) Solve each quadratic equation. Generalized square root property (a) (b)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-12 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-13 A Example 6 Using the Quadratic Formula (Real Solutions) (cont.) Solution set:

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-14 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-15 A Example 8 Solving a Three-Part Inequality (page 429) Solve. Add 8. Divide by 6. Solution set:

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-16 Graphs of Equations B The Rectangular Coordinate System ▪ The Pythagorean Theorem and the Distance Formula ▪ The Midpoint Formula ▪ Graphing Equations ▪ Circles

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-17 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-18 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-19 B Example 3(a) Finding Ordered Pairs That are Solutions to Equations (page 434) x –1–1 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-20 Find three ordered pairs that are solutions to. B Example 3(b) Finding Ordered Pairs That are Solutions to Equations (page 434) x –2–2 –1–1 2 Three ordered pairs that are solutions are (−2, −3), (−1, 0), and (2, −3).

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-21 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.)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-22 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-23 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)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-24 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-25 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-26 B Example 6(a) Graphing Circles (page 437) Graph Write the equation as (h, k) = (1, –2), r = 3

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-27 B Example 6(b) Graphing Circles (page 437) Graph (h, k) = (0, 0), r = 2 Write the equation as

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-28 Functions C Relations and Functions ▪ Domain and Range ▪ Determining Functions from Graphs or Equations ▪ Function Notation ▪ Increasing, Decreasing, and Constant Functions

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-29 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)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-30 C Example 1 Deciding Whether Relations Define Functions (cont.) (b) N is not a function because the x-value –4 has two y-values.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-31 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)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-32 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-33 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}

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-34 C Example 3(b) Finding Domains and Ranges from Graphs (cont.) Give the domain and range of the relation. Domain: Range:

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-35 Give the domain and range of the relation. Domain: [–5, 5] Range: [–3, 3] C Example 3(c) Finding Domains and Ranges from Graphs (cont.)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-36 C Example 3(d) Finding Domains and Ranges from Graphs (cont.) Give the domain and range of the relation. Domain: Range:

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-37 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-38 Determine if the relation is a function and give the domain and range. y is found by squaring x by 2 and adding 3. C Example 4(b) Identifying Functions, Domains, and Ranges (cont.) y = x 2 + 3 Domain: Range: Each value of x corresponds to just one value of y, so the relation is a function.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-39 Determine if the relation is a function and give the domain and 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-40 Determine if the relation is a function and give the domain and range. y is found by dividing 3 by x + 2. C Example 4(d) Identifying Functions, Domains, and Ranges (cont.) Domain: Range: Each value of x corresponds to just one value of y, so the relation is a function.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-41 Let and. Find f(–3), f(r), and g(r + 2). C Example 5 Using Function Notation (page 445)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-42 Find f(–1) for each function. (a) f(x) = 2x 2 – 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-43 Find f(–1) for each function. (c)(d) C Example 6 Using Function Notation (cont.) f(–1) = 5f( – 1) = 0

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-44 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 onDecreasing on Constant on

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-45 Graphing Techniques D Stretching and Shrinking ▪ Reflecting ▪ Symmetry ▪ Translations

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-46 D Example 1 Stretching or Shrinking a Graph (page 451) x –2–2 –1–1 0 1 2 41 1 ¼ 00 1 ¼ 41 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-47 D Example 2 Reflecting a Graph Across an Axis (page 452) x –2–2 –1–1 0 1 2 2–2–2 1–1 00 1 –1–1 2–2 Create a table of values. 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-48 D Example 2 Reflecting a Graph Across an Axis (cont.) x –2–2 –1–1 0 1 2 22 11 00 1 1 22 Create a table of values. 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

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-49 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-50 D Example 3(a) Testing for Symmetry with Respect to an Axis (cont.)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-51 D Example 3(b) 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 the same as the original equation. The graph is symmetric with respect to the y-axis. Replace y with –y: 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-52 D Example 3(b) Testing for Symmetry with Respect to an Axis (cont.)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-53 D Example 3(c) 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 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-54 D Example 3(c) Testing for Symmetry with Respect to an Axis (cont.)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-55 D Example 3(d) 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 the same as the original equation. The graph is symmetric with respect to the y-axis. Replace y with –y: 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-56 D Example 3(d) Testing for Symmetry with Respect to an Axis (cont.)

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-57 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 result is the same as the original equation. The graph is symmetric with respect to the origin.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-58 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 result is not the same as the original equation. The graph is not symmetric with respect to the origin.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-59 D Example 5(a) Translating Graphs (page 455) Graph. Compare a table of values for with. x –2–2 –1–1 0 1 2 46 13 02 13 46 The graph of f(x) is the same as the graph of g(x) translated 2 units up.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-60 D Example 5(b) Translating Graphs (page 455) Graph. x –4–4 –3–3 –2 –2 –1 –1 0 1 164 91 40 11 04 Compare a table of values for with. 19

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-61 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.

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-62 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.

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