SECTION 3.1 Properties of Linear Functions and Linear Models 139 OBJECTIVES 1 Graph Linear Functions (p. 139) 2 Use Average Rate of Change to Identify Linear Functions (p. 139) 3 Determine Whether a Linear Function Is Increasing, Decreasing, or Constant (p. 142) 4 Build Linear Models from Verbal Descriptions (p. 143) 3.1 Properties of Linear Functions and Linear Models Now Work the ‘Are You Prepared?’ problems on page 145. • Lines (Section 1.5, pp. 32–43) • Graphs of Equations in Two Variables; Intercepts; Symmetry (Section 1.3, pp. 20–26) • Solving Equations (Section A.6, pp. A44–A55) • Functions (Section 2.1, pp. 61–73) • The Graph of a Function (Section 2.2, pp. 77–81) • Properties of Functions (Section 2.3, pp. 87–95) PREPARING FOR THIS SECTION Before getting started, review the following: Graphing a Linear Function Graph the linear function f x x3 7. ( ) = − + What are the domain and the range of f ? Solution EXAMPLE 1 This is a linear function with slope m 3 = − and y -intercept b 7. = To graph this function, plot the point 0, 7 , ( ) the y -intercept, and use the slope to find an additional point by moving right 1 unit and down 3 units. See Figure 1.The domain and the range of f are each the set of all real numbers. 1 Graph Linear Functions In Section 1.5 we discussed lines. In particular, for nonvertical lines we developed the slope–intercept form of the equation of a line, y mx b. = + When the slope–intercept form of a line is written using function notation, the result is a linear function . DEFINITION Linear Function A linear function is a function of the form f x mx b ( ) = + The graph of a linear function is a line with slope m and y -intercept b. Its domain is the set of all real numbers. Functions that are not linear are said to be nonlinear . John Urschel John Urschel was selected in the fifth round of the 2014 NFL draft by the Baltimore Ravens. After his third season in the NFL, John retired from professional football. After retiring from football, John began working on his PhD in mathematics at Massachusetts Institute of Technology. He earned his PhD in 2021. He is now an assistant professor in the MIT Math department. Credit: Tribune Content Agency LLC/Alamy Stock Photo Alternatively, an additional point can be found by evaluating the function at some x 0. ≠ For x f 1, 1 3 1 7 4 ( ) = = − ⋅ + = , so the point 1, 4 ( ) lies on the graph. Now Work PROBLEMS 13(a) AND (b) 2 Use Average Rate of Change to Identify Linear Functions Look at Table 1 on the next page, which shows several values of the independent variable x and corresponding values of the dependent variable y for the function f x x3 7. ( ) = − + Notice that as the value of the independent variable x increases by 1, the value of the dependent variable y decreases by 3. That is, the average rate of change of y with respect to x is a constant, 3. − Figure 1 ( ) =− + f x x3 7 x Dy 5 23 Dx 5 1 y 1 3 5 1 3 5 (1, 4) (0, 7)
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