Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 6 ≤ x ≤ 18.

x	f(x)
12	39
18	29
24	19
30	9

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \( [a, b] \) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 6 \) and \( b = 18 \).

Step2: Identify \( f(6) \) and \( f(18) \)

From the table, when \( x = 6 \), \( f(6)=49 \); when \( x = 18 \), \( f(18)=29 \).

Step3: Substitute into the formula

Substitute \( a = 6 \), \( b = 18 \), \( f(6)=49 \), and \( f(18)=29 \) into the formula:

$$ \frac{f(18) - f(6)}{18 - 6}=\frac{29 - 49}{18 - 6} $$

Step4: Simplify the numerator and denominator

First, calculate the numerator: \( 29 - 49=-20 \).
Then, calculate the denominator: \( 18 - 6 = 12 \).
So we have \( \frac{-20}{12} \).

Step5: Reduce the fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 4:

$$ \frac{-20\div4}{12\div4}=\frac{-5}{3} $$

Answer:

\( -\frac{5}{3} \)