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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 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 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.

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.