Question

What is the equation of the normal line of `f(x)=-4x-10` at `x=-2`?

Answer 1

`y+2=1/4(x+2)`
or
`y=x/4-3/2`

Explanation:

The normal line at `x=-2` is the line perpendicular to the tangent line at this point. The slope of the tangent line is simply the derivative evaluated at `x=-2`.

`f^'(-2)=-4`

The slope of the line normal to the tangent line is the negative reciprocal of this i.e. `m=1/4`. Now we have a slope, but we still need a point to write the line. So, evaluate the function at `x=-2`.

`f(-2)=-4(-2)-10=-2`

We now have the slope and a point, so use point slope form to acquire the equation of the normal line.

`y+2=1/4(x+2)`
or
`y=x/4-3/2`