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#