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 2 )?

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 \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 1\) and \(b=2\).

Step2: Find \(f(1)\) and \(f(2)\) from the graph

From the graph, when \(x = 1\), \(y=f(1)=5\) (since the point \((1,5)\) lies on the graph of \(y = f(x)\)), and when \(x = 2\), \(y = f(2)=7\) (since the point \((2,7)\) lies on the graph of \(y = f(x)\)).

Step3: Substitute values into the formula

Substitute \(a = 1\), \(b = 2\), \(f(1)=5\), and \(f(2)=7\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(2)-f(1)}{2 - 1}=\frac{7 - 5}{2-1}\).

Step4: Simplify the expression

\(\frac{7 - 5}{2-1}=\frac{2}{1}=2\).

Answer:

\(2\)