Normal Line

 

Videos

Normal Line Example 1

Normal Line Example 1

Transcript Hi, everyone! Welcome back to integralcalc.com. Today we’re going to be doing another problem with the normal line. And in this one, we’ve been given the function y = x^2 and we’ve been asked to find the normal line of this curve at the point (-2,4). So the normal line is interesting. We know about ... Read More

Normal Line Example 2

Normal Line Example 2

Transcript Hi, everyone! Welcome back to integralcalc.com. Today we’re going to be talking about how to find the normal line to the graph y = 2x^2 + 3x – 5 at the point (2,9). So just to talk about the normal line briefly, we’ve got a graph and I’ve gone ahead and graphed it here. The ... Read More

Normal Line Example 3

Normal Line Example 3

Transcript Hi, everyone! Welcome back to integralcalc.com. Today we’re going to be talking about how we’re going to find the normal line to a graph. And in this one, we’ve been given the function y = 5 – x – 2x^2. And we’ve been asked to find the equation of the normal line to the graph ... Read More

     Subscribe to me on YouTube

 

Equation of the Normal Line

The normal line is the line which is perpendicular to the tangent line. Remember that the slopes of perpendicular lines are negative reciprocals of one another. Therefore, the slope of the normal line is the negative reciprocal of the slope of the tangent line.

This is really convenient, because we know that the slope of the tangent line is the derivative of the original function. Therefore, we know already that the slope of the normal line is -1 divided by the derivative of the original function.

As an example, if the slope of the tangent line is 17, then the slope of the normal line, as its inverse reciprocal, is -\frac{1}{17}.

If we’re given a point on the graph at which we’re supposed to find the normal line to the curve, and given what we know now about the slope of the normal line, we should be able to find the equation of the normal line. After all, if we’re given a point that falls on the normal line (the point where the original function, its tangent line, and the normal line all intersect), and if we can calculate the slope of the normal line by finding the inverse reciprocal of the slope of the tangent line at the intersection point, then we can write the equation of the normal line.

The steps to find the equation of the normal line are the following.

  1. Find the derivative of the original function. In other words, given f(x), find f'(x).
  2. Plug the point at which we’re supposed to find the equation of the normal line into f'(x). Plugging the point into the derivative will give us the slope of the tangent line at that specific point on the graph. This will be the slope of the tangent line. If we needed to, we could use this slope and the point we were given to find the equation of the tangent line. But we don’t need to in order to find the equation of the normal line, so for now, we’ll skip it.
  3. Take the inverse reciprocal of the slope of the tangent line at that point. The inverse reciprocal of the slope of the tangent line will be the slope of the normal line. Remember that, to take the inverse reciprocal of a number, we just find -1 divided by that number.
  4. Now that we have the slope of the normal line, and a point on the normal line (the point where the original function, its tangent line, and the normal line all intersect), we can use these two pieces of information to find the equation of the normal line. We’ll use the point-slope form of the equation of a line since we have a point and a slope. The equation of the line in point-slope form is y-y_1=m(x-x_1), where m is the slope and (x_1,y_1) is the point where the original function, its tangent line, and normal line all intersect one another. Plug your point and your slope into this equation and then simplify the equation as much as possible. This is the equation of the normal line.

 

© 2012 integralCALC
Privacy Policy