Question

a function is graphed below. on which interval of x is the average rate of change of the function the greatest? answer x = 0 to x = 2 x = 2 to x = 6 x = 6 to x = 12 x = 12 to x = 24

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Calculate the average rate of change for \(x = 0\) to \(x=2\)

Here \(a = 0\), \(b = 2\), \(f(0)=0\), \(f(2) = 4\).

$$ \frac{f(2)-f(0)}{2-0}=\frac{4 - 0}{2}=\frac{4}{2}=2 $$

Step3: Calculate the average rate of change for \(x = 2\) to \(x = 6\)

Here \(a = 2\), \(b = 6\), \(f(2)=4\), \(f(6)=10\).

$$ \frac{f(6)-f(2)}{6 - 2}=\frac{10 - 4}{4}=\frac{6}{4}=1.5 $$

Step4: Calculate the average rate of change for \(x = 6\) to \(x = 12\)

Here \(a = 6\), \(b = 12\), \(f(6)=10\), \(f(12)=22\).

$$ \frac{f(12)-f(6)}{12 - 6}=\frac{22-10}{6}=\frac{12}{6}=2 $$

Step5: Calculate the average rate of change for \(x = 12\) to \(x = 24\)

Here \(a = 12\), \(b = 24\), \(f(12)=22\), \(f(24)=31\).

$$ \frac{f(24)-f(12)}{24 - 12}=\frac{31-22}{12}=\frac{9}{12}=0.75 $$

Answer:

\(x = 0\) to \(x = 2\) and \(x = 6\) to \(x = 12\) (since both have an average - rate - of - change of \(2\) which is greater than \(1.5\) and \(0.75\))