Introduction to Graphing Section 1.1 Introduction to Graphing Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives 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.

Cartesian Coordinate System

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)

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.

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.

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

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).

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).

Example Graph 2x + 3y = 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.

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

Example Graph y = x2 – 9x – 12 . Make a table of values. (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.

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

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

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

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

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.

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.