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

We can find the slope of a line between two points as long as we know the coordinates of both points. Since these two points lie on a function whose equation we know, we can use the function to determine the coordinates of the points.

When x = 2 we have f(2) = (2)^{2} = 4, so the coordinates of the first point are (2, 4).

When x = 4 we have f(4) = (4)^{2} = 16 so the coordinates of the second point are (4, 16).

Now that we know the coordinates of both points we can calculate the rise and run:

We have

rise = 16 – 4 = 12

and

run = 4 – 2 = 2.

The slope of the line between these two points is

To think of this in a more formula-oriented way, we have a = 2 and b = 4, so

f(a) = 4 and f(b) = 16.

Then the slope of the line between the two points is

Example 2

If f(x) = x^{3} + 1, find the slope of the secant line between the two points on the function where x = 1 and x = 3.

We have

Now we've got all the ingredients to find the slope of the line between these points:

Example 3

Let f(x) = 1 – x^{2}. What is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)) where a = -1 and h = -0.25.

f(-1) = 0, so our starting point is (-1, 0).

We have
a + h = -1 + (-0.25) = -1.25, so f(a + h) = f(-1.25) = -0.5625, and our ending point is
(-1.25, -0.5625).
This picture shows our starting and ending points and the secant line between them:

To find the slope of the secant line, we use our beautiful formula, which is finding "rise over run":