238 CHAPTER 2 Graphs and Functions Lines can be graphed by finding ordered pairs and plotting them. Although only two points are necessary to graph a linear function, we usually plot a third point as a check. The intercepts are often good points to choose for graphing lines. 2.4 Linear Functions ■ Basic Concepts of Linear Functions ■ Standard Form Ax +By =C ■ Slope ■ Average Rate of Change ■ Linear Models Basic Concepts of Linear Functions We begin our study of specific functions by looking at linear functions. Linear Function A function ƒ is a linear function if, for real numbers a and b, ƒ1x2 =ax +b. If a≠0, then the domain and the range of ƒ are both 1-∞, ∞2. EXAMPLE 1 Graphing a Linear Function Using Intercepts Graph ƒ1x2 = -2x + 6. Give the domain and range. SOLUTION The x-intercept is found by letting ƒ1x2 = 0 and solving for x. ƒ1x2 = -2x + 6 0 = -2x + 6 Let ƒ1x2 = 0. x = 3 Add 2x. Divide by 2. We plot the x-intercept 13, 02. The y-intercept is found by evaluating ƒ102. ƒ1x2 = -2x + 6 ƒ102 = -2102 + 6 Let x = 0. ƒ102 = 6 Multiply, and then add. Therefore, another point on the graph is the y-intercept, 10, 62. We plot this point and join the two points with a straight-line graph. We use the point 12, 22 as a check. See Figure 31. The domain and the range are both 1-∞, ∞2. The corresponding calculator graph with ƒ1x2 = y1 is shown in Figure 32. y1 = −2x+6 −2.5 −10 10.5 10 Figure 32 x y f(x) = –2x + 6 (3, 0) 0 x-intercept y-intercept (0, 6) Check point (2, 2) Figure 31 S Now Try Exercise 13.
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