# What is the equation of the line normal to  f(x)=2(x-1)^2-2x+4 at  x=-3?

##### 1 Answer
May 25, 2017

$y = \frac{x}{18} + 41 + \frac{5}{6}$

#### Explanation:

$f \left(x\right) = 2 {x}^{2} - 6 x + 6$

$f ' \left(x\right) = 4 x - 6$

$f ' \left(- 3\right) = - 12 - 6 = - 18$

We want a line $y = \frac{1}{18} x + b$, where $\left(- 3 , f \left(- 3\right)\right)$ is one of its points.

$f \left(- 3\right) = 2 \cdot 9 + 18 + 6 = 42$

$42 = \frac{1}{18} \left(- 3\right) + b$

$b = 42 - \frac{1}{6} = 41 + \frac{5}{6}$