Question

question 17
1 part -- of 1 point

find the average rate of change for the function between the given values.
\\(f(x) = 3x - 6\\); from 1 to 2

a. -3
b. -6
c. 6
d. 3

Explanation:

Identify the given function and interval

We are given the linear function \(f(x) = 3x - 6\) and need to find its average rate of change over the interval from \(x_1 = 1\) to \(x_2 = 2\).

Recall the average rate of change formula

Using the Average Rate of Change knowledge point, the formula to find the average rate of change of a function \(f(x)\) over an interval \([a, b]\) is:

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

Here, we set \(a = 1\) and \(b = 2\).

Evaluate the function at the boundaries

We evaluate the function at the endpoints of the interval:

• For \(a = 1\):

$$f(1) = 3(1) - 6 = 3 - 6 = -3$$

• For \(b = 2\):

$$f(2) = 3(2) - 6 = 6 - 6 = 0$$

Calculate the average rate of change

Substitute the evaluated values into the formula:

$$\text{Average Rate of Change} = \frac{f(2) - f(1)}{2 - 1} = \frac{0 - (-3)}{1} = \frac{3}{1} = 3$$

Relate to linear function properties

For any linear function of the form \(f(x) = mx + b\), the rate of change is constant and always equal to the slope \(m\). Since our function is \(f(x) = 3x - 6\), the slope is \(3\), which matches our calculated average rate of change. This corresponds to option D.

Answer:

• (A) -3
• (B) -6
• (C) 6
• (D) 3 (Correct answer)