346 CHAPTER 6 Algebra, Graphs, and Functions Example 7 Writing an Equation in Slope–Intercept Form a) Write x y 4 3 9 − = in slope–intercept form. b) Graph the equation. Solution a) To write x y 4 3 9 − = in slope–intercept form, we solve the given equation for y. x y x x y x y x y x y x y x 4 3 9 4 4 3 4 9 3 4 9 3 3 4 9 3 4 3 9 3 or 4 3 3 − = − − = − + − = − + − − = − + − = − − + − = − Thus, in slope–intercept form, the equation is y x 3. 4 3 = − b) The y-intercept is (0, 3), − and the slope is . 4 3 Plot a point at (0, 3) − on the y-axis, then move up 4 units and to the right 3 units to obtain the second point (see Fig. 6.16). Draw a line through the two points. 7 Now try Exercise 77 3 2 1 4 21 22 25 y x 4x 2 3y 5 9 22 23 24 25 21 1 2 3 4 5 23 24 Figure 6.16 y 4 5 25 3 2 21 22 23 24 2221 2423 25 1 2 3 x 1 4 Figure 6.17 Example 8 Determining the Equation of a Line from Its Graph Determine the equation of the line in Fig. 6.17. Solution If we determine the slope and the y-intercept of the line, we can write the equation using slope–intercept form, y mx b. = + We see from the graph that the y-intercept is (0, 1); thus, b 1. = The slope of the line is negative because the graph falls from left to right. The change in y is 1 unit for every 3-unit change in x. Thus, m, the slope of the line, is . 1 3− = + = − + y mx b y x 1 3 1 The equation of the line is y x 1 3 1. = − + 7 Now try Exercise 79 y 1 3 22 24 21 23 1 2 3 4 x 21 y 5 2 2 Figure 6.18 Example 9 Horizontal and Vertical Lines In the Cartesian coordinate system, graph (a) y 2 = and (b) x 3. = − Solution a) For any value of x, the value of y is 2. Therefore, the graph will be a horizontal line through y 2 = (Fig. 6.18). b) For any value of y, the value of x is 3. − Therefore, the graph will be a vertical line through x 3 = − (Fig. 6.19). Note that the graph of y 2 = has a slope of 0. The slope of the graph of x 3 = − is undefined. 7 Now try Exercise 83 y 2 1 22 24 21 2625 1 2 x 21 22 x 5 23 23 Figure 6.19 In graphing the equations in this section, we labeled the horizontal axis the x-axis and the vertical axis the y-axis. For each equation, we can determine values for y by substituting values for x. Since the value of y depends on the value of x, we refer to y as the dependent variable and x as the independent variable. We label the vertical
RkJQdWJsaXNoZXIy NjM5ODQ=