255 2.5 Equations of Lines and Linear Models NOTE Generalizing from Example 3, we see that the slope m of the graph of the equation Ax + By = C is - A B , and the y-intercept is A0, C BB. EXAMPLE 4 Using the Slope-Intercept Form (GivenTwo Points) Write an equation of the line through the points 11, 12 and 12, 42. Then graph the line using the slope-intercept form. SOLUTION In Example 2, we used the point-slope form in a similar problem. Here we show an alternative method using the slope-intercept form. First, find the slope. m= 4 - 1 2 - 1 = 3 1 = 3 Definition of slope Now substitute 3 for m in y = mx + b and choose one of the given points, say 11, 12, to find the value of b. y = mx + b Slope-intercept form 1 = 3112 + b m= 3, x = 1, y = 1 The y-intercept is 10, b2. b = -2 Solve for b. The slope-intercept form is y = 3x - 2. The graph is shown in Figure 46. We can plot 10, -22 and then use the definition of slope to arrive at 11, 12. Verify that 12, 42 also lies on the line. S Now Try Exercise 19. (1, 1) (2, 4) (0, –2) y = 3x – 2 y changes 3 units. x changes 1 unit. x 0 y Figure 46 EXAMPLE 5 Finding an Equation from a Graph Use the graph of the linear function ƒ shown in Figure 47 to complete the following. (a) Identify the slope, y-intercept, and x-intercept. (b) Write an equation that defines ƒ. SOLUTION (a) The line falls 1 unit each time the x-value increases 3 units. Therefore, the slope is -1 3 = - 1 3 . The graph intersects the y-axis at the y-intercept 10, -12 and the x-axis at the x-intercept 1-3, 02. (b) The slope is m= - 1 3 , and the y-intercept is 10, -12. y = ƒ1x2 = mx + b Slope-intercept form ƒ1x2 = - 1 3 x - 1 m= - 1 3 , b = -1 S Now Try Exercise 45. y y = f(x) x 0 3 1 –1 –3 Figure 47
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