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# Derivatives

# Slope of a Line Between Two Points on a Function

We've already done this in the case where the function is a line. What happens if the line isn't straight? What if it's snake-shaped or U-shaped? What if it's as curvy as a mudflap? Now we'll find the slope of a line between two points on any wonky function we like.

### Sample Problem

Find the slope of the line between the two points shown.

Since we're given a formula for the function *f*, we can say

Then we can find the rise and run from this picture:

A line between two points on a function is called a **secant line**.

Asking to find the slope of the "secant line" between two points on a function means the same thing as asking to find the slope of the "line" between those two points.

A secant line is a line between two points on a function. We've been thinking about a secant line as a line that starts at the point on *f* where *x* = *a*, and ends at the point on *f* where *x* = *b*. For what's coming, it will be helpful to think of starting at (*a*, *f*(*a*)) and ending at some "other point." The "other point" will be described by how far away its *x*-value is from *a*. Poor *b* is getting fired, to be replaced by *a* + *h*.

*h* is whatever we need to make *a* + *h* equal the number formerly known as *b*, not to be confused with the artist formally known as Prince.

### Sample Problem

Before, we had *a* = 1 and *b* = 1.5. *a* and *b* were equally important:

Now we have *a* = 1 and *h* = 0.5. *a* is important, and the distance of the other point from *a* is important:

### Sample Problem

Before, we had *a* = 1 and *b* = -2:

Now we have *a* = 1 and *h* = -3:

The moral of the story is that we can think of a secant line on the function *f* as a line that starts at the point on *f* where *x* = *a* and ends at the point on *f* where *x* = *a* + *h*:

Since *h* measures the distance (and direction) from our starting point to our ending point, *h* is conveniently equal to the "run" we need for the slope formula:

We find a lovely formula for the slope of the secant line on the function *f* from *x* = *a* to *x* =* a *+ *h*:

If *h* is negative the formula will be the same, and the picture will look slightly different: