Digital Lesson Graphs of Equations 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 2(3) – (–1) = 7 7=7 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Original Equation Substitute for x and y. Equation is satisfied. 2 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. 3. Connect the points using straight lines or smooth curves. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Example: Sketch the graph of y = –2x + 3. 1. Find several solution points of the equation. x –2 –1 0 1 2 y = –2x + 3 (x, y) y = –2(–2) + 3 = 7 (–2, 7) y = –2(–1) + 3 = 5 (–1, 5) y = –2(0) + 3 = 3 (0, 3) y = –2(1) + 3 = 1 (1, 1) y = –2(2) + 3 = –1 (2, –1) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Sketch the graph of y = –2x + 3. 2. Plot the points in the coordinate plane. x y (x, y) –2 –1 7 5 (–2, 7) (–1, 5) 0 3 (0, 3) 1 2 1 –1 (1, 1) (2, –1) y 8 4 x 4 4 8 –4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Sketch the graph of y = –2x + 3. 3. Connect the points with a straight line. y 8 4 x 4 4 8 –4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Sketch the graph of y = (x – 1)2. x –2 –1 0 1 2 y 9 4 1 0 1 (x, y) (–2, 9) (–1, 4) (0, 1) (1, 0) (2, 1) 3 4 4 9 (3, 4) (4, 9) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y 8 6 2 x –2 2 4 7 Example: Sketch the graph of y = | x | + 1. y x y (x, y) –2 –1 0 1 2 3 2 1 2 3 (–2, 3) (–1, 2) (0, 1) (1, 2) (2, 3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 2 x –2 2 8 The points at which the graph intersects the the xor 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Procedure for finding the x- and y- intercepts of the graph of an equation algebraically: 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Find the x- and y-intercepts of the graph of y = x2 + 4x – 5. To find the x-intercepts, let y = 0 and solve for x. 0 = x2 + 4x – 5 0 = (x – 1)(x + 5) x–1=0 x=1 Substitute 0 for y. Factor. x + 5 = 0 Set each factor equal to 0. 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 = 02 + 4(0) – 5 = –5 So, the y-intercept is (0, –5). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Procedure for finding the x- and y-intercepts of the graph of an equation graphically: To find the x-intercepts of the graph of an equation, locate the points at which the graph intersects the x-axis. To find the y-intercepts of the graph of an equation, locate the points at which the graph intersects the y-axis. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Find the x- and y-intercepts of the graph of x = | y | – 2 shown below. y The graph intersects the x-axis at (–2, 0). The graph intersects the y-axis at (0, 2) and at (0, –2). 2 x –3 1 2 3 The x-intercept is (–2, 0). The y-intercepts are (0, 2) and (0, –2). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13