Problem 18 - Algebra and Trigonometry

Question

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ g(x)=5+\frac{1}{2} x ; \quad x=1, x=5 $$

Step-by-Step Solution

Answer

The average rate of change is 0.5.

Step 1: Understand the Problem

We need to calculate the average rate of change of the function \( g(x) = 5 + \frac{1}{2} x \) between \( x = 1 \) and \( x = 5 \). The average rate of change of a function between two points is essentially the slope of the line connecting those points on the graph of the function.

Step 2: Evaluate the Function at x = 1

Substitute \( x = 1 \) into the function \( g(x) = 5 + \frac{1}{2} x \): \[g(1) = 5 + \frac{1}{2} \times 1 = 5 + 0.5 = 5.5\]

Step 3: Evaluate the Function at x = 5

Substitute \( x = 5 \) into the function \( g(x) = 5 + \frac{1}{2} x \): \[g(5) = 5 + \frac{1}{2} \times 5 = 5 + 2.5 = 7.5\]

Step 4: Calculate the Average Rate of Change

The average rate of change is given by the formula:\[\frac{g(5) - g(1)}{5 - 1}\]Substitute the values from previous steps:\[\frac{7.5 - 5.5}{5 - 1} = \frac{2}{4} = 0.5\]

Key Concepts

Linear Functions

Linear functions are equations that create a straight line when graphed. They are expressed in the form \( f(x) = mx + b \), where \( m \) represents the slope, and \( b \) stands for the y-intercept. These lines intuitively describe a constant rate of change. For instance, the function \( g(x) = 5 + \frac{1}{2}x \) is linear. The term \( \frac{1}{2} \) in this function is the slope, indicating how steep the line is, while 5 is the y-intercept, showing where the line crosses the y-axis.
Linear functions are straightforward because of their predictable pattern. Every unit increase in \( x \) results in a consistent increase in \( g(x) \) by the value of the slope, \( \frac{1}{2} \) in our example. This consistency is why linear functions are favorites for predicting future values, especially in economics and science.

Function Evaluation

Function evaluation is the process of finding the output of a function for specific values of its variable. Simply put, it's about plugging in a number for \( x \) and finding \( f(x) \).
For example, to evaluate \( g(x) = 5 + \frac{1}{2}x \) at \( x = 1 \), you substitute 1 in place of \( x \). You calculate:

• \( g(1) = 5 + \frac{1}{2} \times 1 \)
• \( = 5 + 0.5 \)
• \( = 5.5 \)

Evaluating a function helps us understand the behavior and values of a function at specific points. This technique is essential in understanding how one variable affects another, providing a clearer picture of the function’s performance under various conditions.

Slope Calculation

Calculating the slope is essential for understanding the rate of change between two points on a graph. The slope tells us how much y increases as x increases. The formula for calculating the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

For the function \( g(x) = 5 + \frac{1}{2}x \), we calculate the average rate of change between \( x = 1 \) and \( x = 5 \).
Previously, we found:

• \( g(1) = 5.5 \)
• \( g(5) = 7.5 \)

Now, use the slope formula:

• \( m = \frac{7.5 - 5.5}{5 - 1} \)
• \( = \frac{2}{4} \)
• \( = 0.5 \)

This slope of 0.5 indicates that for each increase by 1 in \( x \), \( g(x) \) increases by 0.5. Understanding slope calculation helps us interpret changes and trends in data effectively.