390 CHAPTER 6 Algebra, Graphs, and Functions The graphs of linear functions are straight lines that will pass the vertical line test. In Section 6.6, we discussed how to graph linear equations. Linear functions can be graphed by plotting points, by using intercepts, or by using the slope and y-intercept. Solution a) We substitute 68,000 for x in the function. = + = + = + = s x x s ( ) 200 0.06 (68,000) 200 0.06(68,000) 200 4080 4280 Thus, if Claudia sells one truck for $68,000 in a given week, her salary for that week will be $4280. b) In the function, replace s x( ) with the weekly salary, 1580, and solve for x. = + = + = = s x x x x x ( ) 200 0.06 1580 200 0.06 1380 0.06 23,000 Thus, if Claudia’s weekly salary was $1580, her weekly sales were $23,000. 7 Now try Exercise 21 Example 4 Graphing a Linear Function Graph = − + f x x ( ) 2 3 by using the slope and y-intercept. Solution We can rewrite this function as = − + y x2 3. From Section 6.6, we know that the slope is −2 and the y-intercept is (0, 3). Plot (0, 3) on the y-axis. Then plot the next point by moving down 2 units and to the right 1 unit (see Fig. 6.44). A third point has been plotted in the same way. The graph of = − + f x x ( ) 2 3 is the line drawn through these three points. 7 Now try Exercise 45 y 5 3 2 1 22 23 24 2221 2423 3 4 5 6 x y 5 22x 1 3 21 1 2 4 Figure 6.44 Quadratic Functions and Their Graphs The standard form of a quadratic equation is = + + ≠ y ax bx c a, 0. 2 We will learn shortly that graphs of equations of this form always pass the vertical line test and are functions. Therefore, equations of the form = + + ≠ y ax bx c a, 0, 2 may be referred to as quadratic functions. We may express quadratic functions using function notation as = + + f x ax bx c ( ) . 2 Two examples of quadratic functions are = + − y x x 2 5 7 2 and f x x ( ) 4. 1 2 2 = − + Example 5 Landing on the Moon On July 20, 1969, Neil Armstrong became the first person to walk on the moon. The velocity, v, of his spacecraft, the Eagle, in meters per second, was a function of time before touchdown, t, given by = = + v f t t ( ) 3.2 0.45 The height of the spacecraft, h, above the moon’s surface, in meters, was also a function of time before touchdown, given by = = + h g t t t ( ) 1.6 0.45 2 What was the velocity of the spacecraft and its height above the moon’s surface a) at 3 seconds before touchdown? b) at touchdown (0 seconds)? m Neil Armstrong on the moon NASA Pictures/Alamy Stock Photo
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