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 average rate of change formula

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

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

From the graph, when \(x=-1\), \(y=-1\) (so \(f(-1)=-1\)), and when \(x = 1\), \(y = 1\) (so \(f(1)=1\)).

Step3: Substitute into the formula

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

Step4: Simplify the expression

\(\frac{1 + 1}{1+1}=\frac{2}{2}=1\).

Answer:

\(1\)