Question 154214
{{{f(x) = x^2 - x}}} Start with the given equation.



{{{f(3) = (3)^2 - 3}}} Plug in {{{x=3}}}



{{{f(3) = 9 - 3}}} Square 3 to get 9



{{{f(3) = 6}}} Subtract



So the point (3,6) is on the graph



----------


{{{f(x) = x^2 - x}}} Start with the given equation.



{{{f(5) = (5)^2 - 5}}} Plug in {{{x=5}}}



{{{f(5) = 25 - 5}}} Square 5 to get 25



{{{f(5) = 20}}} Subtract



So the point (5,20) is on the graph



So the average rate of change from x=3 to x=5 is simply the slope of the line that goes through the points (3,6) and (5,20)



So let's find the slope of the line that goes through the points (3,6) and (5,20)



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(20-6)/(5-3)}}} Plug in {{{y[2]=20}}}, {{{y[1]=6}}}, {{{x[2]=5}}}, and {{{x[1]=3}}} 



{{{m=(14)/(5-3)}}} Subtract {{{6}}} from {{{20}}} to get {{{14}}}



{{{m=(14)/(2)}}} Subtract {{{3}}} from {{{5}}} to get {{{2}}}



{{{m=7}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(3,6\right)] and *[Tex \LARGE \left(5,20\right)] is {{{m=7}}}


Since the slope is 7, this means that the average rate of change from x=3 to x=5 is 7