Get the answer to "What is the equation of the line that..."

Zane Decker

Answered question

2022-04-08

What is the equation of the line that is normal to

f (x) = {(x + 2)}^{2} - 5 x + 2

at x=3?

Quimpoah3b

Beginner2022-04-09Added 13 answers

Explanation:
Require to differentiate so expand bracket in f(x).

f (x) = x^{2} + 4 x + 4 - 5 x + 2 = x^{2} - x + 6

f'(x)=2x-1
Evaluating f'(3) gives the gradient of the tangent and f(3)
the value of the y-coord.
f'(3)=2(3)-1=6-1=5=m

f (3) = {(3 + 2)}^{2} - 5 (3) + 2 = 25 - 15 + 2 = 12 \to (3, 12)

The product of gradients of perpendicular lines is -1
hence m of normal

= - \frac{1}{5}

equation of normal:

y - b = m (x - a), m = - \frac{1}{5}

and (a,b)=(3, 12)

y - 12 = - \frac{1}{5} (x - 3)

(multiply both sides by 5 to eliminate fraction)

5 y - 60 = - x + 3 \to 5 y + x - 63 = 0

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