Question

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

Explanation:

Step1: Recall the formula for average rate of change

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

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

From the graph, when \(x=-1\), \(f(-1)=-20\) (since the point \((-1,-20)\) lies on the graph of \(y = f(x)\)), and when \(x = 1\), \(f(1)=40\) (since the point \((1,40)\) lies on the graph of \(y = f(x)\)).

Step3: Substitute values 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

\(\frac{60}{2}=30\).

Answer:

\(30\)