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Chapter 1 | Functions and Graphs
a. Describe the distance D (in miles) Jessica runs as a linear function of her run time t (in minutes). b. Sketch a graph of D . c. Interpret the meaning of the slope. Solution a. At time t =0, Jessica is at her house, so D (0) =0. At time t =78 minutes, Jessica has finished running 9 mi, so D (78) =9. The slope of the linear function is m = 9−0 78−0 = 3 26 . The y -intercept is (0, 0), so the equation for this linear function is D ( t ) = 3 26 t . b. To graph D , use the fact that the graph passes through the origin and has slope m =3/26.
c. The slope m = 3/26 ≈ 0.115 describes the distance (in miles) Jessica runs per minute, or her average velocity.
Polynomials A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form (1.7) f ( x ) = a n x n + a n −1 x n −1 +…+ a 1 x + a 0 for some integer n ≥0 and constants a n , a n −1 ,…, a 0 , where a n ≠0. In the case when n =0, we allow for a 0 =0; if a 0 =0, the function f ( x ) =0 is called the zero function . The value n is called the degree of the polynomial; the constant a n is called the leading coefficient . A linear function of the form f ( x ) = mx + b is a polynomial of degree 1 if m ≠0 and degree 0 if m =0. A polynomial of degree 0 is also called a constant function . A polynomial function of degree 2 is called a quadratic function . In particular, a quadratic function has the form f ( x ) = ax 2 + bx + c , where a ≠0. A polynomial function of degree 3 is called a cubic function . Power Functions Some polynomial functions are power functions. A power function is any function of the form f ( x ) = ax b , where a and b are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then f ( x ) = ax n is a polynomial. If n is even, then f ( x ) = ax n is an even function because f (− x ) = a (− x ) n = ax n if n is even. If n is odd, then f ( x ) = ax n is an odd function because f (− x ) = a (− x ) n =− ax n if n is odd ( Figure 1.18 ).
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