How do you find the equation of a normal line to a curve at a given point?

1 Answer
Aug 1, 2014

The equation of a normal line will have the form

y = mx + b

and its slope will be the negative reciprocal of the curve's derivative at the point. That is to say, take the value of the derivative at the point, divide 1 by it, and then multiply that value by -1. You then solve for b after plugging in the x and y coordinates of the point, as well as m.

This is much better illustrated with an example:

Let's say that we are expected to find the equation of a line normal to the curve f(x) = x^2 at the point (2, 4). A normal line is a line perpendicular to the tangent line, so we will take the derivative of f(x) to find the slope of the tangent line, and then take the negative reciprocal of this slope, to find the slope of the normal line.

d/dx f(x) = 2x

d/dx f(2) = 2*2 = 4

The negative reciprocal of 4 is -1/4. We now have a value for m:

y = -1/4 x + b

The last step is to plug in the coordinates of our point and solve for b:

4 = -1/4 * 2 + b

4 = -2/4 + b

4 + 1/2 = b

b = 9/2

Now we have everything needed to put our full equation together:

y = -1/4 x + 9/2