Textbook Question
Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. |y| = -x
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2. Graphs of Equations
Two-Variable Equations
2:40 minutesProblem 7
Textbook 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.
Verified step by step guidance1
Step 1: Understand the problem. The car's value decreases by $4000 each year, starting from $36,000. This is a linear depreciation problem.
Step 2: Write the formula for the car's value. The formula for linear depreciation is given by \( y = mx + b \), where \( m \) is the rate of depreciation and \( b \) is the initial value. Here, \( m = -4000 \) and \( b = 36000 \). So, the formula is \( y = -4000x + 36000 \).
Step 3: Use the formula to find when the car's value is $12,000. Set \( y = 12000 \) in the equation \( y = -4000x + 36000 \) and solve for \( x \).
Step 4: Rearrange the equation \( 12000 = -4000x + 36000 \) to solve for \( x \). Subtract 36000 from both sides to get \( -24000 = -4000x \).
Step 5: Divide both sides by -4000 to find \( x \). This will give you the number of years after which the car's value will be $12,000.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 linearly over time, making it essential to understand how to formulate and manipulate linear equations to model the depreciation.
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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's value decreases by a fixed amount each year, which is a key aspect of understanding how to create the formula for the car's value over time and to calculate when it reaches a specific value.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate system to visualize the relationship between the variables. For this problem, it is important to graph the formula derived in part (a) to illustrate the car's value over time and to identify the point where the value reaches $12,000, providing a visual representation of the solution.
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