College Algebra

Suppose $f(x)=x^2-7$ and we want to compute the average rate of change from $x=1$ to $x=1+h\text{.}$ We still follow the same procedure we always use for average rate of change. Both $1$ and $1+h$ are $x$-values, so we need to plug them into our function to get the $y$-values:

In the last calculation, we mulitplied out the square and combined like terms. The reason is that we will now be cominbing it with other expressions, so simplifying now makes our next step a little bit easier. Now that we know the $y$-values, we just need to compute the slope between the points $(1,f(1))$ and $(1+h,f(1+h))\text{,}$ which we now know is $(1,-6)$ and $(1+h,2h+h^2-6)\text{.}$ Parentheses are going to be very important for this step! Remember that with slope we can subtract in either order, as long as we are consistent on top and bottom. In this case, we recommend starting with the one that has the h’s to make the simplifying a little bit easier:

Therefore, our answer is that the average rate of change of $f(x)$ on $[1,1+h]$ is $2+h\text{.}$

Suppose $f(x)=2x^2-1$ and we want to compute the average rate of change of $f(x)$ on $[a,a+h]\text{.}$ We still follow the same procedure we always use for average rate of change. Both $a$ and $a+h$ are $x$-values, so we need to plug them into our function to get the $y$-values:

In the last calculation, we mulitplied out the square and combined like terms. The reason is that we will now be cominbing it with other expressions, so simplifying now makes our next step a little bit easier. Now that we know the $y$-values, we just need to compute the slope between the points $(a,f(a))$ and $(a+h,f(a+h))\text{,}$ which we now know is $(a,2a^2-1)$ and $(a+h,2a^2+4ah+2h^2-1)\text{.}$ Parentheses are going to be very important for this step! Remember that with slope we can subtract in either order, as long as we are consistent on top and bottom. In this case, we recommend starting with the one that has the h’s to make the simplifying a little bit easier:

Therefore, our answer is that the average rate of change of $f(x)$ on $[a,a+h]$ is $4a+2h\text{.}$