Question

the function ( y = f(x) ) is graphed below. what is the average rate of change of the function ( f(x) ) on the interval ( -6leq xleq -1 )?

Explanation:

Step1: Identify 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, \(a=-6\) and \(b = - 1\).

Step2: Find \(f(-6)\) and \(f(-1)\) from the graph

From the graph, when \(x=-6\), \(f(-6)=-5\) (assuming the \(y\) - value at \(x = - 6\) is \(-5\) based on the grid - like structure of the graph). When \(x=-1\), \(f(-1)=25\) (assuming the \(y\) - value at \(x=-1\) is \(25\) based on the grid - like structure of the graph).

Step3: Substitute values into the formula

Substitute \(a=-6\), \(b = - 1\), \(f(a)=f(-6)=-5\) and \(f(b)=f(-1)=25\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{25-(-5)}{-1-(-6)}=\frac{25 + 5}{-1 + 6}=\frac{30}{5}\).

Answer:

\(6\)