Graph Skills: Unit Eight

Determining the Slope of a Curve at the Point of Tangency

In the unit on Slope, we talked about measuring the slope of a straight line. Now we will discuss how to find the slope of a point on a curve. One of the differences between the slope of a straight line and the slope of a curve is that the slope of a straight line is constant, while the slope of a curve changes from point to point.

As you should recall, to find the slope of a line you need to:

Identify two points on the line.
Select one to be (x1, y1) and the other to be (x2, y2).
Use the equation:

For a quick example and review of how to calculate the slope of a straight line, click on the button below.

Now let's use the slope formula in a nonlinear relationship. Let's try using the procedure outlined above to find the slope of the curve shown below.

From point A (0, 2) to point B (1, 2.5)

From point B (1, 2.5) to point C (2, 4)

From point C (2, 4) to point D (3, 8)

Here we see that the slope of the curve changes as you move along it. For this reason, we measure the slope of a curve at just one point. For example, instead of measuring the slope as the change between any two points (between A and B or B and C), we measure the slope of the curve at a single point (at A or C).

Tangent Line
To do this we must introduce the concept of a tangent. A tangent is a straight line that touches a curve at a single point and does not cross through it. The point where the curve and the tangent meet is called the point of tangency. Both of the figures below show a tangent line to the curve.

This curve has a tangent line to the curve with point A being the point of tangency. In this case, the slope of the tangent line is positive.

This curve has a tangent line to the curve with point A being the point of tangency. In this case, the slope of the tangent line is negative.

The line on this graph crosses the curve in two places. This line is not tangent to the curve.

The slope of a curve at a point is equal to the slope of the straight line that is tangent to the curve at that point.

Example

What is the slope of the curve at point A?

The slope of the curve at point A is equal to the slope of the straight line BC. By finding the slope of the straight line BC, we have found the slope of the curve at point A.

The slope at point A is 1/2, or .5.

This is the slope of the curve only at point A. To find the slope of the curve at any other point, we would need to draw a tangent line at that point and then determine the slope of that tangent line.

You are now ready to try a practice problem. If you have already completed the first practice problem for this unit you may wish to try the additional practice.