Finding the Normal Line to a Curve: Definition & Equation

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  • 0:02 Normal Line to a Curve Defined
  • 0:41 The Problem
  • 1:44 Taking the Derivative
  • 3:00 Finding the Normal Line
  • 3:36 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

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.

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