Is the average of two endpoint's instantaneous rates of change always the same as the average rate of change of the interval on a parabola? Is the average of two endpoint's instantaneous rates of change always the same as the average rate of change of the interval on a parabola? If not, which additional circumstances would cause it to always be the same?
EDIT: How would you verify this algebraically? Answered
EDIT 2: They both result in the same answer, but do they get their answer from the same line that touches/intercepts the graph, or do they get their answer from different lines (secant, tangent, etc.) that touch/intercept the graph, and because the two lines are always parallel to each other, they result in the same answer?

A parabola is the graph of a quadratic polynomial $f(x)=ax^2+bx+c.$ The average rate of change of $f$ on the interval $[r,s]$ is $\frac{f(s)-f(r)}{s-r}.$ You can check that this computes to $\frac{a(s^2-r^2)+b(s-r)}{s-r} = a(s+r)+b.$ The instantaneous rate of change of a function is its derivative $f'$. In this case $f'(x)=2ax+b,$ so the instantaneous rates at the endpoints are $f'(s)=2as+b$ and $f'(r)=2ar+b.$ The average of these two values is easily seen to be $a(s+r)+b,$ which is what you wanted.
