A luxury car rental company charges 120 per day in addition...

A luxury car rental company charges $\$ 120$ per day in addition to a flat fee of $\$ 65$ for insurance. (See Example 1.) a. Write an equation that represents the cost $y$ (in dollars) to rent the car for $x$ days. b. Graph the equation. c. What is the $y$ -intercept and what does it mean in the context of this problem? d. Use the model from part (a) to determine the cost of driving the rental car for 2 days, 5 days, and 7 days. e. If the company offers a weekly rate of $\$ 799$, is this a better deal than paying by the day for 7 days? f. Find the total cost of driving the car for 4 days if the sales tax is $6 \% .$ g. Is it reasonable to use negative values of $x$ in the equation from part (a)? Why or why not?

A luxury car rental company charges $\$ 120$ per day in addition to a flat fee of $\$ 65$ for insurance. (See Example 1.)
a. Write an equation that represents the cost $y$ (in dollars) to rent the car for $x$ days.
b. Graph the equation.
c. What is the $y$ -intercept and what does it mean in the context of this problem?
d. Use the model from part (a) to determine the cost of driving the rental car for 2 days, 5 days, and 7 days.
e. If the company offers a weekly rate of $\$ 799$, is this a better deal than paying by the day for 7 days?
f. Find the total cost of driving the car for 4 days if the sales tax is $6 \% .$
g. Is it reasonable to use negative values of $x$ in the equation from part (a)? Why or why not?

Key Concepts

Domain of Functions

The domain of a function refers to the set of all possible input values (typically the independent variable) for which the function is defined. In real-world contexts like rental durations, this concept helps to determine the meaningful and practical values (such as non-negative days) that can be used in the model.

Sales Tax Calculations

Sales tax calculations involve applying a percentage rate to a total amount to determine the extra cost imposed by tax regulations. This concept is essential when adjusting the calculated total cost to account for additional charges, ensuring that the final amount reflects all applicable fees.

Cost Modeling

Cost modeling involves creating mathematical representations to estimate the total cost based on given pricing structures. These models typically combine fixed costs with variable costs, and are useful in planning and comparing expenses over different time periods or quantities.

Comparing Pricing Options

Comparing pricing options involves evaluating different cost structures to determine which is more economical under certain conditions. This concept is important when decisions involve choices between varying pricing models, such as a daily rate versus a weekly rate, by analyzing their respective total costs for a given period.

Y-intercept

The y-intercept is the value of y when x equals zero. In the context of cost modeling, it often represents a fixed fee or an initial cost that is incurred regardless of the number of units (such as days) being considered. Understanding the y-intercept helps in interpreting the starting cost independent of the variable component.

Linear Functions

A linear function is a mathematical model that describes a relationship between two variables with a constant rate of change. It is typically expressed in the form y = mx + b, where m represents the slope (or rate of change) and b represents the y-intercept (or initial value when x is zero). This concept is widely used in algebra to model real-world situations where there is a fixed starting amount and a regular increment added for each unit increase in another variable.

Graphing Linear Equations

Graphing is the process of plotting a linear equation on a coordinate plane, allowing one to visually interpret the relationship between the variables. This involves identifying the slope and y-intercept from the equation and drawing a line that shows how costs or other values change as the other variable increases.

Slope

The slope in a linear function indicates how much y increases (or decreases) for each unit increase in x. In cost-related problems, the slope represents the variable cost per unit, such as a per-day rental cost, making it a crucial concept for understanding the rate at which expenses accumulate over time.

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