# What is the equation of the normal line of f(x)=x^2-x+5 at x=2?

Dec 31, 2016

$y = \frac{23 - x}{3}$

#### Explanation:

Differentiate:
$f p r i m e \left(x\right) = 2 x - 1$
$f p r i m e \left(2\right) = 2 \left(2\right) - 1$
$= 3$
Therefore the normal has gradient $- \frac{1}{3}$ because "${m}_{1} {m}_{2} = - 1$"for perpendicular lines.
Therefore the equation of the normal line is $y = - \frac{x}{3} + c$.
This has to pass through the point $\left(1 , f \left(2\right)\right)$, that is, $\left(1 , 7\right)$.
Therefore $7 = - \frac{2}{3} + c$ hence $c = \frac{23}{3}$.
So the line is $y = - \frac{x}{3} + \frac{23}{3}$.