Problem 42 - College Algebra with Modeling and Visualization

Question

Exercises \(39-44\) : Average Rate of Change Find the average rate of change of \(f\) from \(-2\) to \(2 .\) What is the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\), where \(x_{1} \neq x_{2} ?\) $$ f(x)=\frac{5}{3} x $$

Step-by-Step Solution

Answer

The average rate of change is \(\frac{5}{3}\).

Step 1: Understand the Problem

We need to find the average rate of change of the function \(f(x) = \frac{5}{3}x\) from \(-2\) to \(2\). The average rate of change is similar to the slope between these two points on the graph.

Step 2: Find the Function Values

Calculate \(f(x)\) for \(x = -2\) and \(x = 2\).For \(x = -2\): \[ f(-2) = \frac{5}{3}(-2) = -\frac{10}{3} \]For \(x = 2\): \[ f(2) = \frac{5}{3}(2) = \frac{10}{3} \]

Step 3: Use the Average Rate of Change Formula

The average rate of change of \(f\) from \(x_1\) to \(x_2\) is given by:\[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]Here, \(x_1 = -2\) and \(x_2 = 2\), so substitute these values:\[ \frac{f(2) - f(-2)}{2 - (-2)} \]

Step 4: Substitute the Function Values

Substitute \(f(2) = \frac{10}{3}\) and \(f(-2) = -\frac{10}{3}\):\[ \frac{\frac{10}{3} - (-\frac{10}{3})}{2 - (-2)} = \frac{\frac{10}{3} + \frac{10}{3}}{4} \]

Step 5: Simplify the Expression

Simplify the numerator and divide by the denominator:\[ \frac{\frac{20}{3}}{4} = \frac{20}{3} \times \frac{1}{4} = \frac{20}{12} = \frac{5}{3} \]

Step 6: Conclusion

Thus, the average rate of change of the function \(f(x) = \frac{5}{3}x\) from \(-2\) to \(2\) is \(\frac{5}{3}\).

Key Concepts

Average Rate of Change

To understand the average rate of change, think about how much something changes, on average, over a specific interval. It's like figuring out how fast a car is going over a trip, not just at one moment. Mathematically, the average rate of change of a function from one point to another is essentially the slope of the line that connects these two points on a graph.
For the function \( f(x) = \frac{5}{3}x \), calculating the average rate of change between \( x = -2 \) and \( x = 2 \) involves these steps:

• Calculate the function values at both \( x = -2 \) and \( x = 2 \).
• Use these values in the formula \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).

By understanding this concept, you're essentially measuring how steep or flat the line is between your chosen points.

Linear Functions

Linear functions are the simplest type of function you can encounter in algebra. They graph as straight lines on a coordinate grid. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In the function \( f(x) = \frac{5}{3}x \), notice how it fits the form of a linear function with \( m = \frac{5}{3} \) and \( b = 0 \). This means:

• The function line crosses the y-axis at the origin (0,0).
• For every unit increase in \( x \), \( f(x) \) increases by \( \frac{5}{3} \) units.

Understanding linear functions is crucial, as they form the basis for much more complex algebraic concepts later on.

Slope Calculation

The slope of a line tells you how steep or flat the line is. Calculating the slope is a fundamental algebra skill, especially for analyzing linear functions. The slope is found using the formula \( \frac{\text{rise}}{\text{run}} \).
In our example with \( f(x) = \frac{5}{3}x \), the slope is already given as the coefficient of \( x \). This means that:

• For every 3 units you move horizontally (the run), the line moves 5 units vertically (the rise).
• This relationship is why the slope is \( \frac{5}{3} \).

Knowing how to calculate and understand slope allows you to describe the behavior of linear functions effectively, making it easier to predict values and understand graphs.