How to Find Average Rate of Change for Functions

Understanding the average rate of change is essential in mathematics, especially when comparing different functions. The average rate of change of a function between two points is calculated as the slope of the line connecting those points. In this example, we will determine the average rate of change for the exponential function y = 4^x and the quadratic function y = 4x^2 over the interval [0, 1].

### Step 1: Calculate y-values for y = 4^x

1. **Find y when x = 0:**
- y = 4^0 = 1.

2. **Find y when x = 1:**
- y = 4^1 = 4.

Now we have two points for the exponential function: (0, 1) and (1, 4).

### Step 2: Calculate y-values for y = 4x^2

1. **Find y when x = 0:**
- y = 4·0^2 = 0.

2. **Find y when x = 1:**
- y = 4·1^2 = 4.

Now we have two points for the quadratic function: (0, 0) and (1, 4).

### Step 3: Apply the Average Rate of Change Formula
The formula for the average rate of change is:

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

Where \(f(a)\) and \(f(b)\) are the function values at points a and b.

**For y = 4^x:**
- Using the points (0, 1) and (1, 4):
\[
ext{Average Rate of Change} = \frac{4 - 1}{1 - 0} = \frac{3}{1} = 3
\]

**For y = 4x^2:**
- Using the points (0, 0) and (1, 4):
\[
ext{Average Rate of Change} = \frac{4 - 0}{1 - 0} = \frac{4}{1} = 4
\]

### Conclusion
After performing these calculations, we find that the average rate of change for y = 4^x is 3, while for y = 4x^2 it is 4. Therefore, the quadratic function actually has a greater average rate of change over the interval [0, 1]. This process not only helps in understanding how to compare functions but also reinforces fundamental concepts of slopes and rates of change, which are applicable in various real-world scenarios such as physics, economics, and biology where analyzing trends is crucial.