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