We have been given a function [tex]g(x)=x^2+2x-5[/tex]. We are asked to find average rate of change of the function over the interval [tex]-6\leq x\leq 1[/tex].
We will use average rate of change formula to solve our given problem.
[tex]\text{Average rate of change}=\frac{f(b)-f(a)}{b-a}[/tex]
For our given function [tex]b=1[/tex] and [tex]a=-6[/tex].
[tex]\text{Average rate of change}=\frac{g(1)-g(-6)}{1-(-6)}[/tex]
[tex]g(1)=1^2+2(1)-5[/tex]
[tex]g(1)=1+2-5[/tex]
[tex]g(1)=-2[/tex]
[tex]g(-6)=(-6)^2+2(-6)-5[/tex]
[tex]g(-6)=36-12-5[/tex]
[tex]g(-6)=19[/tex]
[tex]\text{Average rate of change}=\frac{-2-19}{1+6}[/tex]
[tex]\text{Average rate of change}=\frac{-21}{7}[/tex]
[tex]\text{Average rate of change}=-3[/tex]
Therefore, the average rate of change of the function is [tex]-3[/tex] over the interval [tex]-6\leq x\leq 1[/tex].
