Question

a function is shown in the table below. on which interval of x is the average rate of change of the function the greatest? answer x = 1 to x = 6 x = 6 to x = 18 x = 18 to x = 34 x = 34 to x = 42

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).

Step2: Calculate the average rate of change for \(x = 1\) to \(x = 6\)

Here, \(x_1=1,y_1 = 6,x_2=6,y_2 = 16\).
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{16 - 6}{6 - 1}=\frac{10}{5}=2\).

Step3: Calculate the average rate of change for \(x = 6\) to \(x = 18\)

Here, \(x_1 = 6,y_1=16,x_2 = 18,y_2=25\).
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{25 - 16}{18 - 6}=\frac{9}{12}=\frac{3}{4}=0.75\).

Step4: Calculate the average rate of change for \(x = 18\) to \(x = 34\)

Here, \(x_1=18,y_1 = 25,x_2=34,y_2=33\).
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{33 - 25}{34 - 18}=\frac{8}{16}=0.5\).

Step5: Calculate the average rate of change for \(x = 34\) to \(x = 42\)

Here, \(x_1=34,y_1 = 33,x_2=42,y_2=36\).
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{36 - 33}{42 - 34}=\frac{3}{8}=0.375\).

Answer:

\(x = 1\) to \(x = 6\)