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Chapter 3: Problem 39

Graph the indicated functions. For a certain model of truck, its resale value \(V\) (in dollars) as a function of its mileage \(m\) is \(V=50,000-0.2 m .\) Plot \(V\) as a function of \(m\) for \(m \leq 100,000\) mi.

Short Answer

Expert verified
Plot a line from (0, 50,000) to (100,000, 30,000) to represent the function.

Step by step solution

01

Understanding the Function

The given function is \( V = 50,000 - 0.2m \), where \( V \) is the resale value in dollars, and \( m \) is the mileage in miles. It is a linear function of the form \( V = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
02

Identifying Important Points

To graph the function, we need to identify key points. The y-intercept is \( V = 50,000 \) when \( m = 0 \). When \( m = 100,000 \), substituting in the function gives \( V = 50,000 - 0.2(100,000) \), which simplifies to \( V = 30,000 \).
03

Drawing the Graph Axes

Draw the x-axis representing mileage \( m \) and the y-axis representing resale value \( V \). Label the y-axis from \( 30,000 \) to \( 50,000 \) and the x-axis from \( 0 \) to \( 100,000 \).
04

Plotting Key Points

Plot the point (0, 50,000) as the y-intercept and the point (100,000, 30,000). These points give you the endpoints of the function's graph.
05

Drawing the Line

Connect the plotted points with a straight line. This line represents the function \( V = 50,000 - 0.2m \) within the domain of \( 0 \leq m \leq 100,000 \). Ensure your line covers this interval on the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Slope
In the context of a linear function, the slope is a crucial element that determines the steepness and direction of the graph. For the truck model's resale value function, the slope is represented by the coefficient of "m," which is \(-0.2\). This slope indicates how much the resale value \(V\) decreases as the mileage \(m\) increases.
  • A negative slope means the graph slants downwards; for every additional mile on the truck, the value decreases by \(0.2\) dollars.
  • Slope is often found in the equation \(y = mx + c\) where \(m\) is the slope.
  • Here, it shows that high mileage has a considerable effect on lowering the resale value, causing a decline.
Understanding this concept is vital as it tells how sensitive the resale value is to additional mileage. This helps in planning and setting expectations for how quickly a truck's value might depreciate over time.
Y-intercept Importance
The y-intercept is another vital part of understanding linear equations. In our truck resale value equation, the y-intercept is \(50,000\). This value represents the resale value of the truck when the mileage \(m\) is zero.
  • In practical terms, it is the hypothetical starting value of the truck before it has been driven any miles.
  • The y-intercept is found at the point where the function crosses the y-axis.
For our function \(V = 50,000 - 0.2m\), it shows what the initial sale price of the truck might be without any wear from usage. Knowing the y-intercept allows owners to predict the baseline value of their asset before considering mileage impact.
Exploring the Mileage Effect
Mileage is a critical factor that impacts many aspects of vehicle value, particularly resale. In the equation \(V = 50,000 - 0.2m\), mileage \(m\) is the independent variable and significantly influences the resale price. As mileage increases, the resale value decreases, indicating wear and usage on the vehicle.
  • A high mileage figure denotes substantial usage, which generally leads to a faster depreciation of value.
  • Understanding how mileage affects value helps in making better buying or selling decisions.
For instance, if you are a potential buyer or a seller, knowing that every additional mile reduces value by \(0.2\) dollars can guide you in negotiations and deciding a vehicle's worth relative to its mileage.
Understanding Resale Value
Resale value can be thought of as the future worth of a vehicle after considering its initial cost, usage, and inherent depreciation. Within the linear function \(V = 50,000 - 0.2m\), the resale value \(V\) changes inversely with mileage \(m\).
  • This function formula allows us to predict how the truck's resale value will decrease over time.
  • It is helpful for current owners to forecast future value loss and can be a strong negotiation tool during sales.
Understanding how to read and interpret these values is essential for making long-term financial decisions regarding vehicle investments. Additionally, knowing your vehicle's depreciation helps in planning timely upgrades or sales to maximize returns.

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