Question

A new car worth $$36,000 is depreciating in value by 4000 $$per year. a. Write a formula that models the car's value, y, in dollars, after x years. b. Use the formula from part (a) to determine after how many years the car's value will be $12,000. c. Graph the formula from part (a) in the first quadrant of a rectangular coordinate system. Then show your solution to part (b) on the graph.

Accepted Answer

For part (a), recognize that the car's value decreases by a fixed amount each year, which is a linear relationship. The general form of a linear equation modeling depreciation is $$y = mx + b$$, where $$m is $$the rate of change (slope) and $$b is $$the initial value (y-intercept). Identify the initial value $$b as$$ $$36,000 ($$the car's worth at year 0) and the rate of depreciation $$m as$$ $$-4000 ($$since the value decreases by $$4000$$ each year). So, the formula becomes $$y = -4000x + 36000. $$For part (b), use the formula $$y = -4000x + 36000$$ and set $$y$$ equal to $$12,000 to $$find the number of years $$x$$ when the car's value reaches $$12,000. $$This gives the equation $$12000 = -4000x + 36000. $$Solve the equation for $$x by $$isolating the variable: subtract $$36000$$ from both sides to get $$12000 - 36000 = -4000x$$, then simplify and divide both sides by $$-4000 to $$find $$x. $$For part (c), plot the linear equation $$y = -4000x + 36000 on a $$coordinate plane with $$x$$ representing years and $$y$$ representing the car's value. The graph will be a straight line starting at $$(0, 36000)$$ and decreasing by $$4000$$ for each increase of 1 in $$x. $$Mark the point where $$y = 12000 on $$the graph to visually show the solution from part (b).

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Key Concepts

Linear Functions and Modeling

Solving Linear Equations

Graphing Linear Equations