Question

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

Answer 1

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`