4.2 The Slope of a Quadratic Function

If you graph a quadratic you will notice that you do not get a straight line. On the other hand, if you were to look at your graph under a microscope, you might think it was a straight line. In the same sense, though the earth is round, as we walk down the street it looks pretty flat to us poor tiny creatures.

If you look at a quadratic function \(f\) at some particular argument, call it \(z\), and very close to \(z\), then \(f\) will look like a straight line. The line f resembles at argument \(z\) is called the tangent line to \(f\) at argument \(z\), and the slope of this tangent line to \(f\) at \(z\) is called the derivative of \(f\) at argument \(z\). This slope is often written as

\[f'(z), \text{or as} \frac{df}{dz} \text{or} \frac{df(z)}{dz}\]

The tangent line to the function \(f\) at a specific argument is the graph of a linear function. That function is called the linear approximation to \(f\) at argument \(z\). Notice that it is a different function from \(f\) and is typically near \(f\) only when evaluated at an argument \(x\) that is near to \(z\).

The same exact words can be used to define the derivative of any function, \(f\), that looks like a straight line in some vicinity of argument \(z\). \(f\)'s derivative at argument \(z\), which we write as \(f'(z)\) or \(\frac{df(z)}{dz}\), will be the slope of that straight line.

The derivative and tangent line mathlet allows you to enter any function you can construct into it, and look at the graph of its values, and its slopes, that is, its derivative on any interval you choose.

We will next see how to find the derivative of a quadratic function, or any polynomial function given its formula.