How to Calculate the Average Rate of Change in Functions

Calculating the average rate of change of a function is a fundamental concept in mathematics, especially in algebra and calculus. It measures how much a function's output changes, on average, as the input changes over a specific interval. In this example, we are given the function f(x) = -x² + 12 and a table of values to work with. Let's break down the process step by step.

### Step 1: Understanding the Average Rate of Change
The average rate of change of a function between two points, say a and b, can be found using the formula:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

This formula essentially calculates the slope of the line that connects the two points on the function's graph.

### Step 2: Identify the Interval
In our scenario, we need to calculate the average rate of change over the interval [2, 4]. Here, a = 2 and b = 4. We will use the values from the function table provided:
- For x = 2, f(2) = 8
- For x = 4, f(4) = -4

### Step 3: Plugging in Values
Now, we will plug these values into our formula:
- Change in y (f(b) - f(a)): f(4) - f(2) = -4 - 8 = -12
- Change in x (b - a): 4 - 2 = 2

So, substituting these values into the formula gives us:

\[ \text{Average Rate of Change} = \frac{-12}{2} = -6 \]

### Conclusion
Thus, the average rate of change of the function f(x) over the interval [2, 4] is -6. This result indicates that, on average, the function decreases by 6 units for every 1 unit increase in x over this interval.

### Why Only the Values at x = 2 and x = 4?
You might wonder why we only needed the values at x = 2 and x = 4, rather than using all the values in the table. The reason is that the average rate of change specifically looks at the overall change between two defined points. While the other values in the table (like f(1) = 11 and f(3) = 3) provide additional information about the function's behavior, they do not influence the average rate of change between x = 2 and x = 4. This principle is crucial in mathematics, especially for understanding linear approximations and the concept of slopes on a graph.

### Real-World Application
Understanding the average rate of change is helpful in various real-world scenarios, such as calculating speed, determining profit changes over time, or analyzing trends in data. It is a skill that provides valuable insights into how different variables interact and change in relation to one another.

By mastering this concept, you will be better equipped to tackle more complex mathematical problems and apply these principles in practical contexts.