Questions: For the function f(x)=3x+2, the average rate of change will be the same for any interval you choose. Why? (If you are stuck, choose two x-values and find the average rate of change between them.)

For the function f(x)=3x+2, the average rate of change will be the same for any interval you choose. Why? (If you are stuck, choose two x-values and find the average rate of change between them.)

Transcript text: 5. For the function $f(x)=3 x+2$, the average rate of change will be the same for any interval you choose. Why? (If you are stuck, choose two $x$-values and find the average rate of change between them.)

Solution

Solution Steps

Step 1: Understanding the Function

The function given is $ f(x) = 3x + 2 $. This is a linear function, which means its graph is a straight line. The slope of this line is constant, which is a key property of linear functions.

Step 2: Calculating the Average Rate of Change

The average rate of change of a function $ f(x) $ over an interval $[a, b]$ is given by the formula:

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

For the function $ f(x) = 3x + 2 $, let's choose two arbitrary points $ a $ and $ b $.

Step 3: Example Calculation

Let's choose $ a = 1 $ and $ b = 3 $.

Calculate $ f(a) $ and $ f(b) $:

\[ f(1) = 3(1) + 2 = 5 \]

\[ f(3) = 3(3) + 2 = 11 \]

Now, calculate the average rate of change:

\[ \frac{f(3) - f(1)}{3 - 1} = \frac{11 - 5}{2} = \frac{6}{2} = 3 \]

Step 4: Generalization

Since the function is linear, the slope (which is the coefficient of $ x $, here 3) is constant. Therefore, the average rate of change will always be 3, regardless of the interval chosen.