A normal line is a line that is perpendicular to a tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent line.

The steps for finding the equation of a normal line is pretty simple, as long as you’ve mastered finding the equation of a tangent line.

Steps for finding the equation of a normal line:

1. Find the normal point

2. Compute the slope of the function at the x coordinate

- The normal point is the same as the tangent point

- Compute the derivative of the function at the x coordinate

3. Compute the slope of the perpendicular line at the x coordinate

- This is the slope of the tangent line

- The slope (
) is the negative reciprocal of the slope ( ) of the tangent line

4. Use the point-slope formula to find the equation of the normal line

Let’s see some examples.

First example (click here):

Find the normal line of at

1. Find the normal point

2. Compute the slope of the function at the x coordinate

3. Compute the slope of the perpendicular line at the x coordinate

4. Use the point-slope formula to find the equation of the normal line

Next example (click here):

Find the normal line of at

1. Find the normal point

It is already given.

2. Compute the slope of the function at the x coordinate

3. Compute the slope of the perpendicular line at the x coordinate

4. Use the point-slope formula to find the equation of the normal line

Last example (click here):

Find the normal line of at

1. Find the normal point

2. Compute the slope of the function at the x coordinate

3. Compute the slope of the perpendicular line at the x coordinate

4. Use point-slope formula to find the equation of the normal line

Finding the equation of a normal line is very similar to finding the equation of a tangent line. Since the steps are similar, make sure you don’t confuse and mix up the definitions of a tangent line and a normal line. For more practice and help with normal lines, check out Symbolab’s Practice.

Until next time,

Leah

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