Problem 53 - College Algebra with Modeling and Visualization

Question

Compute the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\). Round your answer to two decimal places when appropriate. Interpret your result graphically. $$ f(x)=7 x-2, x_{1}=1, \text { and } x_{2}=4 $$

Step-by-Step Solution

Answer

The average rate of change is 7, representing the slope of the line.

Step 1: Understand the Problem

We are asked to compute the average rate of change of the function \( f(x) = 7x - 2 \) from \( x_1 = 1 \) to \( x_2 = 4 \). The average rate of change between two points on the function is similar to finding the slope between these points.

Step 2: Identify the Formula

The formula for average rate of change is: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]. Here, \( x_1 = 1 \) and \( x_2 = 4 \).

Step 3: Calculate \(f(x_1)\) and \(f(x_2)\)

Substitute \( x_1 = 1 \) into the function: \( f(1) = 7 \times 1 - 2 = 5 \). Substitute \( x_2 = 4 \) into the function: \( f(4) = 7 \times 4 - 2 = 26 \).

Step 4: Compute the Average Rate of Change

Use the values from Step 3 in the formula identified in Step 2: \[ \text{Average Rate of Change} = \frac{26 - 5}{4 - 1} = \frac{21}{3} = 7 \].

Step 5: Graphical Interpretation

The result of 7 represents the slope of the line between the points \((1, f(1)) = (1, 5)\) and \((4, f(4)) = (4, 26)\) on the graph of \( f(x) = 7x - 2 \). Since \( f(x) \) is a linear function, this rate of change is constant across the entire graph.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions in mathematics. A linear function is generally expressed in the form \( f(x) = mx + b \). Here, \( m \) represents the slope, and \( b \) represents the y-intercept of the function. Linear functions create straight lines when graphed.

They are called "linear" because they graph as straight lines. This characteristic makes them easy to analyze geometrically. The function \( f(x) = 7x - 2 \) is a linear function where the slope \( m \) is 7, and the y-intercept \( b \) is -2.

Understanding linear functions involves recognizing their constant rate of change, which is defined by the slope. No matter what two points you choose on this function, the rate of change between them will always be the same, illustrating the concept of a consistent slope. This is key when discussing the average rate of change.

Slope

The slope is a crucial concept when discussing linear functions and their graphical representations. In the context of our function \( f(x) = 7x - 2 \), the slope (7) indicates how steep the line is. Slope is essentially the "rise over run" or the change in \( y \) for a given change in \( x \).

Mathematically, slope is often represented as \( m \) and can be calculated using the formula:

• \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)

For our function, between the points \((1, 5)\) and \((4, 26)\), the slope calculation matches this formula.

• \( m = \frac{26 - 5}{4 - 1} = \frac{21}{3} = 7 \)

This tells us that for every unit increase in \( x \), \( f(x) \) increases by 7. This constant increase is the essence of a linear function's slope.

Graphical Interpretation

Graphically interpreting a function helps to visualize mathematical concepts. For a linear function like \( f(x) = 7x - 2 \), the graph is a straight line.

The graph can be drawn by plotting its y-intercept \((-2)\) and then using the slope \(7\) to determine the direction and steepness. Starting from the y-intercept, we can plot another point using the slope: rise by 7 and run by 1.

This produces a straight line passing through all points following this rule. When plotted between points like \((1, 5)\) and \((4, 26)\), it confirms the constant rate of change we calculated. The average rate of change represents the slope of this line on the graph.

• Every point along this line confirms the function's consistent pace, indicating the reliability of linear models in predicting behavior.

Graphical interpretation further strengthens our understanding by visually reinforcing the mathematical calculations involved in finding the average rate of change.