Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-1leq xleq1$?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-1\) and \(b = 1\).

Step2: Find \(f(-1)\) and \(f(1)\) from the graph

From the graph, when \(x=-1\), \(y=f(-1)=-20\) (by observing the \(y\) - coordinate of the point on the graph with \(x =-1\)). When \(x = 1\), \(y=f(1)=40\) (by observing the \(y\) - coordinate of the point on the graph with \(x = 1\)).

Step3: Substitute into the formula

Substitute \(a=-1\), \(b = 1\), \(f(a)=-20\), and \(f(b)=40\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{40-(-20)}{1-(-1)}=\frac{40 + 20}{1 + 1}\).

Step4: Simplify the expression

\(\frac{40+20}{1 + 1}=\frac{60}{2}=30\).

Answer:

\(30\)