A new car worth $45,000 is depreciating in value by $5000 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 $10,000. 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.

Verified step by step guidance

1. insert step 1: Identify the initial value of the car and the rate of depreciation. The car is initially worth $45,000 and depreciates by $5,000 per year.
2. insert step 2: Write the formula for the car's value after x years. This is a linear depreciation model, so the formula is y = 45000 - 5000x.
3. insert step 3: To find out after how many years the car's value will be $10,000, set y = 10000 in the formula and solve for x.
4. insert step 4: Rearrange the equation 10000 = 45000 - 5000x to solve for x.
5. insert step 5: Graph the equation y = 45000 - 5000x in the first quadrant, and mark the point where y = 10000 to show the solution from part (b).

Key Concepts

Linear Functions

A linear function is a mathematical expression that describes a relationship between two variables, typically in the form y = mx + b, where m is the slope and b is the y-intercept. In this context, the car's value decreases at a constant rate, making it a linear function. Understanding linear functions is essential for modeling the depreciation of the car's value over time.

Depreciation

Depreciation refers to the reduction in the value of an asset over time, often due to wear and tear or obsolescence. In this scenario, the car depreciates by a fixed amount of $5,000 each year. Recognizing how depreciation affects the value of the car is crucial for formulating the equation and solving for the time when the car's value reaches a specific amount.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate system to visualize the relationship between the variables. The x-axis typically represents time (years), while the y-axis represents the car's value. Understanding how to graph the equation derived from the depreciation model helps in visually interpreting the car's value over time and identifying specific points, such as when the value reaches $10,000.