# Exploring the Derivative of a Quadratic Function

Geometrically, you can view the derivative of a function at a point as the slope of the tangent line to the graph of the function at that point.
In the app below, move point along the graph of and compare the value of the

*x*-coordinate of with the slope of the tangent line at . Do you notice anything interesting?Considering the displayed values in the table of the app above, can you make a conjecture about what is the derivative of the function ?

Verify your conjecture by calculating the derivative of using the limit of the difference quotient.

## What if the given function is f(x)=-x^2 instead?

The app below works as the previous one, but in this case we have the function .
If you move point along the graph of this new function and compare the value of the

*x*-coordinate of with the slope of the tangent line at , do you notice anything interesting?Considering the displayed values in the table of the app above, can you make a conjecture about what is the derivative of the function ?

Verify your conjecture by calculating the derivative of using the limit of the difference quotient.