38 CHAPTER 1 Graphs Seeing the Concept To see the role that the slope m plays, graph the following lines on the same screen. = = + =− + = + =− + Y Y x Y x Y x Y x 2 2 2 3 2 3 2 1 2 3 4 5 Figure 60 displays the graphs using Desmos. What do you conclude about lines of the form = + y mx 2? Figure 61 y x b 2 = + Figure 60 y mx 2 = + To see the role of the y-intercept b, graph the following lines on the same screen. = = + = − = + = − Y x Y x Y x Y x Y x 2 2 1 2 1 2 4 2 4 1 2 3 4 5 Figure 61 displays the graphs using Desmos. What do you conclude about lines of the form y x b 2 ? = + =m 0 =m 1 =− m 1 =m 3 =− m 3 =b 0 =b 1 =− b 1 =b 4 =− b 4 When the equation of a line is written in slope-intercept form, it is easy to find the slope m and y-intercept b of the line. For example, suppose that the equation of a line is = − + y x2 7 Compare this equation to = + y mx b. = − + y x2 7 ↑ ↑ = + y mx b The slope of this line is −2 and its y-intercept is 7. Now Work PROBLEM 79 Finding the Slope and y-Intercept of a Line Find the slope m and y-intercept b of the equation + = x y 2 4 8. Graph the equation. Solution EXAMPLE 6 To find the slope and y-intercept, write the equation in slope-intercept form by solving for y. + = = − + = − + x y y x y x 2 4 8 4 2 8 1 2 2 = + y mx b The coefficient of x, − 1 2 , is the slope, and the constant, 2, is the y-intercept. To graph the line with y-intercept 2 and with slope − 1 2 , start at the point ( ) 0, 2 and move to the right 2 units and then down 1 unit to the point ( ) 2, 1 . Draw the line through these points. See Figure 62. Figure 62 =− + y x 1 2 2 x y –3 3 (0, 2) (2, 1) 2 1 4 Now Work PROBLEM 85
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