Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $4 \leq x \leq 5$?

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, \(a = 4\) and \(b=5\).

Step2: Find \(f(4)\) and \(f(5)\) from the graph

From the graph, when \(x = 4\), \(f(4)=0\); when \(x = 5\), \(f(5)=- 6\).

Step3: Substitute into the formula

Substitute \(a = 4\), \(b = 5\), \(f(4)=0\) and \(f(5)=-6\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(5)-f(4)}{5 - 4}=\frac{-6-0}{5 - 4}\).

Step4: Simplify the expression

\(\frac{-6-0}{5 - 4}=\frac{-6}{1}=-6\)

Answer:

\(-6\)