SECTION 3.1 Properties of Linear Functions and Linear Models 143 Straight-line Depreciation Book value is the value of an asset that a company uses to create its balance sheet. Some companies depreciate assets using straight-line depreciation so that the value of the asset declines by a fixed amount each year.The amount of the decline depends on the useful life that the company assigns to the asset. Suppose a company just purchased a fleet of new cars for its sales force at a cost of $31,500 per car. The company chooses to depreciate each vehicle using the straight-line method over 7 years. This means that each car will depreciate by $31,500 7 $4500 = per year. (a) Write a linear function that expresses the book value V of each car as a function of its age, x, in years. (b) Graph the linear function. (c) What is the book value of each car after 3 years? (d) Interpret the slope. (e) When will the book value of each car be $9000? Solution EXAMPLE 4 (a) If we let V x( ) represent the value of each car after x years, then V 0( ) represents the original or initial value of each car, so V 0 $31,500. ( ) = The y -intercept of the linear function is $31,500. Because each car depreciates by $4500 per year, the slope of the linear function is 4500. − The linear function that represents the book value V of each car after x years is V x x 4500 31,500 ( ) = − + (b) Figure 4 shows the graph of V. (c) The book value of each car after 3 years is V 3 4500 3 31,500 $18,000 ( ) = − ⋅ + = (d) Since the slope of V x x 4500 31,500 ( ) = − + is 4500, − the average rate of change of the book value is $4500 year. − So for each additional year that passes, the book value of the car decreases by $4500. Figure 4 ( ) =− + V x x 4500 31,500 31,500 27,000 22,500 18,000 13,500 9000 4500 1 2 3 4 5 6 7 Book value ($) Age of vehicle (years) x V 4 Build Linear Models from Verbal Descriptions When the average rate of change of a function is constant, a linear function can model the relation between the two variables. For example, if a recycling company pays $0.52 per pound for aluminum cans, then the relation between the price p paid and the pounds recycled x can be modeled as the linear function p x x 0.52 , ( ) = with slope m 0.52 dollar 1 pound . = In Words A constant rate of change involving two variables indicates a linear function. Modeling with a Linear Function If the average rate of change of a function is a constant m, a linear function f can be used to model the relation between the two variables as follows: f x mx b ( ) = + where b is the value of f at 0; that is, b f 0 . ( ) = In many applications, the domain of the function consists of numbers greater than or equal to zero. In those cases, f 0( ) is usually called the initial value . (continued)
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