Question

questions 2 through 5 refer to the following.
function ( f ) is defined on the interval ( -3,6 ). the graph of ( y = f(x) ) is shown in the ( xy )-plane. function ( g ) is defined by ( g(x)=x + 2 ).
over which of the following intervals is the average rate of change for function ( f ) the greatest?
a ( -3,-2 )
b ( -3,-1 )
c ( -3,3 )
d ( -3,6 )

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Analyze each interval

• For interval \([-3,-2]\):

Let \(a=-3\) and \(b = - 2\). The average rate of change is \(\frac{f(-2)-f(-3)}{-2-(-3)}=\frac{f(-2)-f(-3)}{1}\). From the graph, \(f(-3)=-3\) and \(f(-2) = 0\), so the average rate of change is \(0-(-3)=3\).

• For interval \([-3,-1]\):

Let \(a=-3\) and \(b=-1\). \(\frac{f(-1)-f(-3)}{-1-(-3)}=\frac{f(-1)-f(-3)}{2}\). \(f(-3)=-3\) and \(f(-1)\approx - 0.5\), so \(\frac{-0.5-(-3)}{2}=\frac{2.5}{2}=1.25\).

• For interval \([-3,3]\):

Let \(a = - 3\) and \(b = 3\). \(\frac{f(3)-f(-3)}{3-(-3)}=\frac{f(3)-f(-3)}{6}\). \(f(-3)=-3\) and \(f(3)=2\), so \(\frac{2-(-3)}{6}=\frac{5}{6}\approx0.83\).

• For interval \([-3,6]\):

Let \(a=-3\) and \(b = 6\). \(\frac{f(6)-f(-3)}{6-(-3)}=\frac{f(6)-f(-3)}{9}\). \(f(-3)=-3\) and \(f(6)=4\), so \(\frac{4-(-3)}{9}=\frac{7}{9}\approx0.78\).

Answer:

A. \([-3,-2]\)