The cost of driving a car includes both fixed costs and...

The cost of driving a car includes both fixed costs and mileage costs. Assume that insurance and car payments cost $\$ 350$ per month and gasoline, oil, and routine maintenance cost $\$ 0.29$ per mile. (a) Find a linear function $f$ that gives the annual cost of driving this car $x$ miles. (b) What does the $y$ -intercept on the graph of $f$ represent?

The cost of driving a car includes both fixed costs and mileage costs. Assume that insurance and car payments cost $\$ 350$ per month and gasoline, oil, and routine maintenance cost $\$ 0.29$ per mile.
(a) Find a linear function $f$ that gives the annual cost of driving this car $x$ miles.
(b) What does the $y$ -intercept on the graph of $f$ represent?

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Key Concepts

Linear Function

A linear function is an algebraic expression in which the relationship between two variables is modeled by an equation of the form y = mx + b, where m represents the rate of change (slope) and b represents the initial value (y-intercept). In cost modeling, this function is useful for predicting outcomes based on a constant rate of change.

Fixed Costs

Fixed costs are expenses that do not change with the level of activity, such as the recurring insurance and car payments in this situation. They remain constant regardless of how many miles are driven and form the constant part of the cost function.

Variable Costs

Variable costs depend on the amount of activity, such as the miles driven. These costs change in direct proportion to the activity level, represented by the term with the slope (m) in the linear function.

Y-Intercept Interpretation

The y-intercept in a linear function represents the value of the dependent variable when the independent variable is zero. In a cost function, it reflects the total fixed expenses incurred even when there is no usage, serving as the starting point of the cost model.

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