Chapter 1 Graphs, Functions, and Models

Section 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling 1.5 Linear Equations, Functions, Zeros and Applications 1.6 Solving Linear Inequalities 2

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. 3

Cartesian Coordinate System 4

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. Plot (3, 5). Move 3 units left. Next, we move 5 units up. Plot the point. (–3, 5) 5

Solutions of Equations Equations in two variables have solutions (x, y) that are ordered pairs. Example: 2x + 3y = 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. 6

Examples a. Determine whether the ordered pair (5, 7) is a solution of 2x + 3y = 18. 2(5) + 3(7) ? 18 10 + 21 ? 18 11 = 18 FALSE (5, 7) is not a solution. b. Determine whether the ordered pair (3, 4) is a solution of 2x + 3y = 18. 2(3) + 3(4) ? 18 6 + 12 ? 18 18 = 18 TRUE (3, 4) is a solution. 7

Graphs of Equations To graph an equation is to make a drawing that represents the solutions of that equation. 8

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 2x + 3y = 18. 2x + 3(0) = 18 2x = 18 x = 9 The x-intercept is (9, 0). 9

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 2x + 3y = 18. 2(0) + 3y = 18 3y = 18 y = 6 The y-intercept is (0, 6). 10

Example Graph 2x + 3y = 18. Thus, is a solution. 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. 11

Example (continued) (9, 0) Graph: 2x + 3y = 18. x-intercept: y-intercept: (0, 6) Third point: 12

Example Make a table of values. Graph y = x2 – 9x – 12 . (12, 24) 24 –2 32 26 12 y (10, –2) 10 (5, 32) 5 (4, 32) 4 (2, 26) 2 (0, 12) (1, –2) 1 (3, 24) 3 (x, y) x Make a table of values. 13

The Distance Formula The distance d between any two points (x1, y1) and (x2, y2) is given by 14

Example Find the distance between the points (–2, 2) and (3, 6). 15

Midpoint Formula If the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are 16

Example Find the midpoint of a segment whose endpoints are (4, 2) and (2, 5). 17

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 = r2. 18

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 = r2 [x  3]2 + [y  (7)]2 = 52 (x  3)2 + (y + 7)2 = 25. 19