Question

select the correct answer.

this table represents function (f).

\\(\

$$\begin{array}{|c|c|c|c|c|c|c|c|}\\hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\\\\\hline f(x) & -4.5 & -2 & -0.5 & 0 & -0.5 & -2 & -4.5 \\\\\\hline\\end{array}$$

\\)

if function (g) is a quadratic function that contains the points ((-3, 5)) and ((0, 14)), which statement is true over the interval (-3, 0)?

• the average rate of change of (f) is less than the average rate of change of (g).
• the average rates of change of (f) and (g) cannot be determined from the given information.
• the average rate of change of (f) is the same as the average rate of change of (g).
• the average rate of change of (f) is more than the average rate of change of (g).

Explanation:

Calculate the average rate of change of \(f\)

$$ \text{Average Rate of Change of } f = \frac{f(0) - f(-3)}{0 - (-3)} = \frac{0 - (-4.5)}{3} = \frac{4.5}{3} = 1.5 $$

Calculate the average rate of change of \(g\)

$$ \text{Average Rate of Change of } g = \frac{g(0) - g(-3)}{0 - (-3)} = \frac{14 - 5}{3} = \frac{9}{3} = 3 $$

Compare the average rates of change

$$ 1.5 < 3 \implies \text{The average rate of change of } f \text{ is less than the average rate of change of } g. $$

Answer:

• (A) The average rate of change of \(f\) is less than the average rate of change of \(g\). (Correct answer)
• (B) The average rates of change of \(f\) and \(g\) cannot be determined from the given information.
• (C) The average rate of change of \(f\) is the same as the average rate of change of \(g\).
• (D) The average rate of change of \(f\) is more than the average rate of change of \(g\).