Problem 63 - College Algebra with Modeling and Visualization

Question

Express a function \(f\) with the specified representation. Cost of Driving In 2008 the average cost of driving a new car was about 50 cents per mile. Give symbolic, graphical, and numerical representations of the cost in dollars of driving x miles. For the numerical representation use a table with x = 1, 2, 3, 4, 5, 6.

Step-by-Step Solution

Answer

Symbolically: \( f(x) = 0.50x \). Numerically: Use x = 1 to 6 with costs ranging from $0.50 to $3.00. Graphically: Linear graph.

Step 1: Define the Symbolic Representation

To express the function symbolically, identify the cost per mile. Here, it is 50 cents per mile. Therefore, the cost function in dollars can be defined as \( f(x) = 0.50x \), where \( x \) is the number of miles driven.

Step 2: Create a Numerical Representation Table

Using the function \( f(x) = 0.50x \), create a table for \( x = 1, 2, 3, 4, 5, 6 \):\[\begin{array}{c|c} x & \text{Cost in dollars} \\hline 1 & 0.50 \ 2 & 1.00 \ 3 & 1.50 \ 4 & 2.00 \ 5 & 2.50 \ 6 & 3.00 \\end{array}\]

Step 3: Develop the Graphical Representation

Use the function \( f(x) = 0.50x \) to create a graph:1. Label the x-axis as 'Miles'.2. Label the y-axis as 'Cost in Dollars'.3. Plot the points (1, 0.50), (2, 1.00), (3, 1.50), (4, 2.00), (5, 2.50), and (6, 3.00).4. Connect the points with a straight line, as the relationship is linear.

Key Concepts

Symbolic Representation

In algebra, a symbolic representation of a function gives a formula that represents the relationship between variables. For the cost of driving problem, we symbolize the cost per mile as a mathematical expression. The rate given is 50 cents per mile, which translates to 0.50 dollars per mile. Thus, the function describing this situation is written as:
\( f(x) = 0.50x \).
Here, \( f(x) \) represents the total cost in dollars, and \( x \) is the number of miles driven.

• "\( f(x) = 0.50x \)" is read as "f of x equals 0.50 times x."
• "0.50x" means multiply the number of miles by the cost per mile (0.50 dollars or 50 cents).

This symbolic form allows us to easily calculate the cost for any number of miles driven by simply substituting \( x \) with the desired mileage.

Numerical Representation

A numerical representation involves expressing the relationship between variables using numbers in a structured format, typically a table. For our function \( f(x) = 0.50x \), we create a table listing different values of miles (\( x \)) and their corresponding costs.
Let’s populate the table:

• If \( x = 1 \), then \( f(1) = 0.50 \times 1 = 0.50 \).
• If \( x = 2 \), then \( f(2) = 0.50 \times 2 = 1.00 \).
• If \( x = 3 \), then \( f(3) = 0.50 \times 3 = 1.50 \).
• If \( x = 4 \), then \( f(4) = 0.50 \times 4 = 2.00 \).
• If \( x = 5 \), then \( f(5) = 0.50 \times 5 = 2.50 \).
• If \( x = 6 \), then \( f(6) = 0.50 \times 6 = 3.00 \).

This table gives us concrete examples of costs for different distances, allowing us to see how the cost increases in a predictable manner as the miles increase. Numerical representations are particularly helpful for understanding how the costs change incrementally.

Graphical Representation

Graphical representation converts numerical data into a visual format, usually a chart or graph. This makes it easier to discern patterns or trends. Using our cost function, \( f(x) = 0.50x \), we can develop a graph.
Here’s how:

• The x-axis represents "Miles driven."
• The y-axis is labeled "Cost in Dollars."
• Plot the data points: (1, 0.50), (2, 1.00), (3, 1.50), (4, 2.00), (5, 2.50), and (6, 3.00).
• Connect these points with a straight line since the relationship is linear.

The straight line on the graph illustrates that for every additional mile driven, the cost increases proportionally at a constant rate. Graphs are excellent tools for visualizing the relationship between two variables and understanding how changes in one variable affect the other.