Problem 10 - Algebra and Trigonometry

Question

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 Average Rate of Change

The average rate of change of a function between two values of the variable is calculated as the difference in function values divided by the difference in the variable values. It's similar to finding the slope of a line joining two points on the function.

Step 2: Calculate Function Values

Evaluate the function at the given x-values. Start with the first value, substituting it into the function: \[ g(1) = 5 + \frac{1}{2}(1) = 5 + 0.5 = 5.5 \] Next, substitute the second value: \[ g(5) = 5 + \frac{1}{2}(5) = 5 + 2.5 = 7.5 \]

Step 3: Compute the Difference in Function Values

Subtract the function value at the first x from the function value at the second x: \[ \Delta g = g(5) - g(1) = 7.5 - 5.5 = 2 \]

Step 4: Compute the Difference in x-values

Subtract the first x-value from the second x-value: \[ \Delta x = 5 - 1 = 4 \]

Step 5: Determine the Average Rate of Change

Divide the difference in function values by the difference in x-values: \[ \text{Average rate of change} = \frac{\Delta g}{\Delta x} = \frac{2}{4} = 0.5 \]

Key Concepts

Linear Function

A linear function is a type of mathematical equation that represents a straight line when graphed on a coordinate plane. It can be expressed in the standard form as \( f(x) = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

In this exercise, the function \( g(x) = 5 + \frac{1}{2} x \) is a linear function. The slope \( m = \frac{1}{2} \) indicates the rate of change of \( g(x) \) with respect to \( x \), and \( b = 5 \) is where the line intersects the y-axis. Understanding linear functions helps us easily calculate the average rate of change between any two points along the line.

Slope

Slope is a measure of the steepness and direction of a straight line. It is calculated as the ratio of the change in the y-values (vertical change) to the change in the x-values (horizontal change). The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be written as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]In this exercise, the slope \( \frac{1}{2} \) describes how much \( g(x) \) increases for each additional unit increase in \( x \). A positive slope indicates that as \( x \) increases, \( g(x) \) also increases. The concept of slope is essential in determining the average rate of change as it essentially represents the same calculation between two distinct points on a linear function.

Function Evaluation

Function evaluation involves plugging specific values into a function to calculate the corresponding output. This step is crucial in our exercise, allowing us to find the function values at particular points:

• First, compute \( g(1) \) by replacing \( x \) with 1 in \( g(x) = 5 + \frac{1}{2}x \):\[ g(1) = 5 + \frac{1}{2}(1) = 5.5 \]


• Next, find \( g(5) \) by replacing \( x \) with 5:\[ g(5) = 5 + \frac{1}{2}(5) = 7.5 \]

Understanding function evaluation is essential for calculating the average rate of change, as it provides the specific endpoints or coordinates required for analysis.

Variable Values

In math, variable values refer to the specific numbers that a variable can take. In functions like \( g(x) = 5 + \frac{1}{2}x \), \( x \) is the variable, while specific numerical values of \( x \) (like 1 and 5 in our exercise) are the variable values.

• The average rate of change calculation involves determining the function's change by evaluating it at these specified numbers.
• For this purpose, \( x = 1 \) and \( x = 5 \) are used to compute \( g(1) \) and \( g(5) \), respectively.

By understanding the role of variable values, you can effectively apply function evaluation and compute necessary mathematical solutions like the average rate of change for a given function.