Digital Lesson Graphs of Equations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation. For instance, the point (–1, 3) is on the graph of 2y – x = 7 because the equation is satisfied when –1 is substituted for x and 3 is substituted for y. That is, 2y – x = 7 Original Equation 2(3) – (–1) = 7 Substitute for x and y. 7 = 7 Equation is satisfied. Definition of Graph

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 To sketch the graph of an equation, 1.Find several solution points of the equation by substituting various values for x and solving the equation for y. 2. Plot the points in the coordinate plane. 1.Connect the points using straight lines or smooth curves. Sketching Graphs

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Sketch the graph of y = –2x Find several solution points of the equation. (2, –1)y = –2(2) + 3 = –12 (1, 1)y = –2(1) + 3 = 11 (0, 3)y = –2(0) + 3 = 30 (–1, 5)y = –2(–1) + 3 = 5–1 (–2, 7)y = –2(–2) + 3 = 7–2 (x, y)y = –2x + 3x Example: Sketch Graph (Linear Function)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Sketch the graph of y = –2x Plot the points in the coordinate plane –4 x y (2, –1)–12 (1, 1)11 (0, 3)30 (–1, 5)5–1 (–2, 7)7–2 (x, y)yx Example continued

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Sketch the graph of y = –2x Connect the points with a straight line –4 x y Example continued

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Sketch the graph of y = (x – 1) 2. (4, 9)94 (3, 4)43 (2, 1)12 (1, 0)01 (0, 1)10 (–1, 4)4–1 (–2, 9)9–2 (x, y)yx y x –2 Example: Sketch Graph (Quadratic Function)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Sketch the graph of y = | x | + 1. (2, 3)32 (1, 2)21 (0, 1)10 (–1, 2)2–1 (–2, 3)3–2 (x, y)yx y x – Example: Sketch Graph (Absolute Value Function)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 The points at which the graph intersects the the x- or y-axis are called intercepts. If (x, 0) satisfies an equation, then the point (x, 0) is called an x-intercept of the graph of the equation. If (0, y) satisfies an equation, then the point (0, y) is called a y-intercept of the graph of the equation. Definition of Intercepts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 To find the x-intercepts of the graph of an equation, substitute 0 for y in the equation and solve for x. To find the y-intercepts of the graph of an equation algebraically, substitute 0 for x in the equation and solve for y. Procedure for finding the x- and y- intercepts of the graph of an equation algebraically: Finding Intercepts Algebraically

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Find the x- and y-intercepts of the graph of y = x 2 + 4x – 5. To find the x-intercepts, let y = 0 and solve for x. 0 = x 2 + 4x – 5 Substitute 0 for y. 0 = (x – 1)(x + 5) Factor. x – 1 = 0 x + 5 = 0 Set each factor equal to 0. x = 1 x = –5 Solve for x. So, the x-intercepts are (1, 0) and (–5, 0). To find the y-intercept, let x = 0 and solve for y. y = (0) – 5 = –5 So, the y-intercept is (0, –5). Example: Find Intercepts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 To find the x-intercepts of the graph of an equation, locate the points at which the graph intersects the x-axis. Procedure for finding the x- and y-intercepts of the graph of an equation graphically: To find the y-intercepts of the graph of an equation, locate the points at which the graph intersects the y-axis. Finding Intercepts Graphically

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Find the x- and y-intercepts of the graph of x = | y | – 2 shown below. y x 1 2 –323 The x-intercept is (–2, 0). The y-intercepts are (0, 2) and (0, –2). The graph intersects the x-axis at (–2, 0). The graph intersects the y-axis at (0, 2) and at (0, –2). Example: Find Intercepts