CHAPTER 1 Graphs Functions and Models 1 1

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CHAPTER 1: Graphs, Functions, and Models 1. 1 1. 2 1. 3 1. 4

CHAPTER 1: Graphs, Functions, and Models 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 Introduction to Graphing Functions and Graphs Linear Functions, Slope, and Applications Equations of Lines and Modeling Linear Equations, Functions, Zeros and Applications Solving Linear Inequalities Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

1. 1 Introduction to Graphing Plot points. · Determine whether an ordered pair is

1. 1 Introduction to Graphing Plot points. · Determine whether an ordered pair is a solution of an equation. · Find the x-and y-intercepts of an equation of the form Ax + By = C. · Graph equations. · Find the distance between two points in the plane and find the midpoint of a segment. · Find an equation of a circle with a given center and radius, and given an equation of a circle in standard form, find the center and the radius. · Graph equations of circles. · Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Cartesian Coordinate System Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide

Cartesian Coordinate System Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 4

Example To graph or plot a point, the first coordinate tells us to move

Example To graph or plot a point, the first coordinate tells us to move left or right from the origin. The second coordinate tells us to move up or down. (– 3, 5) Plot ( 3, 5). Move 3 units left. Next, we move 5 units up. Plot the point. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 5

Solutions of Equations in two variables have solutions (x, y) that are ordered pairs.

Solutions of Equations in two variables have solutions (x, y) that are ordered pairs. Example: 2 x + 3 y = 18 When an ordered pair is substituted into the equation, the result is a true equation. The ordered pair has to be a solution of the equation to receive a true statement. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 6

Examples a. Determine whether the ordered pair ( 5, 7) is a solution of

Examples a. Determine whether the ordered pair ( 5, 7) is a solution of 2 x + 3 y = 18. b. Determine whether the ordered pair (3, 4) is a solution of 2 x + 3 y = 18. 2( 5) + 3(7) ? 18 10 + 21 ? 18 11 = 18 FALSE ( 5, 7) is not a solution. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 2(3) + 3(4) ? 18 6 + 12 ? 18 18 = 18 TRUE (3, 4) is a solution. Slide 1. 1 - 7

Graphs of Equations To graph an equation is to make a drawing that represents

Graphs of Equations To graph an equation is to make a drawing that represents the solutions of that equation. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 8

x-Intercept The point at which the graph crosses the x-axis. An x-intercept is a

x-Intercept The point at which the graph crosses the x-axis. An x-intercept is a point (a, 0). To find a, let y = 0 and solve for x. Example: Find the x-intercept of 2 x + 3 y = 18. 2 x + 3(0) = 18 2 x = 18 x=9 The x-intercept is (9, 0). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 9

y-Intercept The point at which the graph crosses the y-axis. A y-intercept is a

y-Intercept The point at which the graph crosses the y-axis. A y-intercept is a point (0, b). To find b, let x = 0 and solve for y. Example: Find the y-intercept of 2 x + 3 y = 18. 2(0) + 3 y = 18 y=6 The y-intercept is (0, 6). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 10

Example Graph 2 x + 3 y = 18. We already found the x-intercept:

Example Graph 2 x + 3 y = 18. We already found the x-intercept: (9, 0) We already found the y-intercept: (0, 6) We find a third solution as a check. If x is replaced with 5, then Thus, is a solution. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 11

Example (continued) Graph: 2 x + 3 y = 18. x-intercept: (9, 0) y-intercept:

Example (continued) Graph: 2 x + 3 y = 18. x-intercept: (9, 0) y-intercept: (0, 6) Third point: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 12

Example Graph y = x 2 – 9 x – 12. Make a table

Example Graph y = x 2 – 9 x – 12. Make a table of values. x 3 1 0 2 4 5 10 12 y 24 – 2 12 26 32 32 – 2 24 (x, y) ( 3, 24) ( 1, – 2) (0, 12) (2, 26) (4, 32) (5, 32) (10, – 2) (12, 24) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 13

The Distance Formula The distance d between any two points (x 1, y 1)

The Distance Formula The distance d between any two points (x 1, y 1) and (x 2, y 2) is given by Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 14

Example Find the distance between the points (– 2, 2) and (3, 6). Copyright

Example Find the distance between the points (– 2, 2) and (3, 6). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 15

Midpoint Formula If the endpoints of a segment are (x 1, y 1) and

Midpoint Formula If the endpoints of a segment are (x 1, y 1) and (x 2, y 2), then the coordinates of the midpoint are Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 16

Example Find the midpoint of a segment whose endpoints are ( 4, 2) and

Example Find the midpoint of a segment whose endpoints are ( 4, 2) and (2, 5). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 17

Circles A circle is the set of all points in a plane that are

Circles A circle is the set of all points in a plane that are a fixed distance r from a center (h, k). The equation of a circle with center (h, k) and radius r, in standard form, is (x h)2 + (y k)2 = r 2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 18

Example Find an equation of a circle having radius 5 and center (3, 7).

Example Find an equation of a circle having radius 5 and center (3, 7). Using the standard form, we have (x h)2 + (y k)2 = r 2 [x 3]2 + [y ( 7)]2 = 52 (x 3)2 + (y + 7)2 = 25. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1. 1 - 19