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A linear function is a special kind of mathematical relationship between two variables that creates a straight line when graphed. It can be written in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In our exercise, the linear function provided is \(f(x) = -2x + 15\). Here, the slope is \(-2\), meaning that for every unit increase in \(x\), the value of the function decreases by 2. The y-intercept is 15, which is the value of \(f(x)\) when \(x\) is zero.

The slope tells us how quickly or slowly the function is changing. In linear functions, the rate of change remains constant, signifying that the function grows or declines at a steady rate. This constant change is crucial when determining the average rate of change over different intervals.

An interval refers to a range of numbers between two values, often used to describe the domain of a function. In the context of our exercise, intervals represent specific portions of the x-values over which we need to find the average rate of change. The intervals given in our exercise are \(0, 3\), \(1, 5\), \(1, 3\), and \(1, 6\).

When working with intervals, you only consider the values of the function at the endpoints of these intervals. This means evaluating the function at the starting point \(x_1\) and the ending point \(x_2\) for each interval. This is important for calculating the average rate of change, as you only need the values of the function on these endpoints, simplifying the process.

To find the average rate of change, we use the formula \(\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\). This formula helps us determine how much the function changes over a specific interval. For linear functions, it essentially calculates the slope over the given interval.

• First, substitute \(x_1\) and \(x_2\) into the function \(f(x)\) to find \(f(x_1)\) and \(f(x_2)\).
• Next, subtract these results to find the change in function values: \(f(x_{2}) - f(x_{1})\).
• Finally, divide this difference by the change in \(x\): \(x_{2} - x_{1}\).

This consistent method ensures you correctly calculate the average change in the function over varying lengths of \(x\). By applying this process to each interval, you can easily compare how the function behaves differently over each range of x-values.