When you are given a secant line on a graph with ONE point , you can find the other point by using the difference quotients...

How do you find the difference quotients?

Simple!

First look at the function.

When we give this function a point it will look like this

We don't know exact numbers so we will give the x value x and the y value f(x) because it is a function. As you can see this is the point for the first point of the secant line. We need to be able to find the slope for this line in order to move on with anything else.

We need to know the point on the graph to be able to use this formula. We can get the other ordered pair on the graph by adding in another variable.

H stands for the length that we are missing. We use H to show that x has changed. When we write the next x value it will look like this...

We have to add x+h because that shows that we are adding both segments. The second point isn't just H. It is both x and h combined together and that is why we add them. The same thing for the y value. Now that we have our points, we are able to plug it in the slope formula.

Next we are able to cancel variables like x's at the bottom

As a result, we have our difference quotients. Taking the derivative of the quotients formula will get you the slope from the secant line.

f'(x)=2x will be the slope. From here you can use y=mx+b the find the tangent to any point on the function.