Slope of a Line Between Two Points on a Function

We've already done this in the case where the function is a line, but 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:

By letting h get smaller and smaller, we can get a better idea of what the slope of the function is at a particular point, x = a.