Finding the Normal Line to a Curve

Finding the Normal Line to a Curve: Definition & Equation

Finding the normal line to a curve requires several steps, but they aren't hard. In this lesson, learn what the steps are and what each step does so you can better understand the process.

Normal Line to a Curve Defined

The normal line to a curve is the line that is perpendicular to the tangent of the curve at a particular point. For example, for the curve y = x², to draw the normal line to the curve at the point (1, 1), we would first draw the line tangent to the curve at that point. Then we would draw a line perpendicular to that line. It would look like this.

The red line in the graph is the normal line to the curve y = x² at the point (1, 1). Depending on where you are on the curve, the normal line will be different since the tangent line is different.

Now that we know what the normal line to a curve is, let's see what the mathematical process is to find it.

The Problem

Let's look at a problem to show you the process of the curve y = x²+3x at the point where x = -3. Before we begin, we need to ask ourselves, what bits of information do we need to find the normal line? We need to know the point and the slope of the normal line. So when we find these bits of information, we need to make sure to mark them and remember them.

The problem mentions the point where x = -3. What does y equal at that point? We can find that out by plugging x into the curve and solving for y.

We get y = 0 at that point, so our point is (-3, 0). Good. We have one bit of information we need. Now, we need to find our tangent line. To do this, recall that the tangent is how much the curve is changing at that point or the derivative of the curve.

Taking the Derivative

Now, we need to take the derivative to find our tangent. Remember that in order to the draw the normal line, we first had to draw the tangent line. It's the same when working the problem mathematically. So, let's take the derivative of the curve to see what we get.

We need to follow the rules of derivatives to get our derivative. This equation allows us to find the slope at any point of the graph. Our point is (-3, 0), so we'll plug in x = -3 to our derivative in order to find our slope y' at that point. This slope is the slope of the tangent line at that point.

Since our normal line is perpendicular to our tangent line, the slope of the normal line will be the negative reciprocal of the slope of the tangent line. Our tangent line has a slope of -3, so the slope of our normal line is then 1/3. This is the other bit of information we need. Now, we can go about finding the equation for the normal line of the curve at our point.

Finding the Normal Line

We have a slope and a point, so we can use our point-slope form of a line to find our equation. Let's plug in our slope of the normal line, 1/3, and our point, (-3, 0), and solve for y.

Lesson Summary

Let's review. The normal line to a curve is the line that is perpendicular to the tangent of the curve at a particular point. For example, to draw the normal line to the curve at the point, we would first draw the line tangent to the curve at that point. Then we would draw a line perpendicular to that line. To find the normal line, we need to know its point and slope. Once we know those bits of information, we can use our point-slope form of a line to find our equation.