# How do you find the equation for the parabola with the given Vertex (4,-1), point (-2,35)?

Aug 24, 2015

Find the equation of the parabola

Ans: y = 9x^2 - 72x + 143

#### Explanation:

Equation of the parabola : $y = a {x}^{2} + b x + c$. Find a, b, and c
Vertex (4, -1)
x-coordinate of vertex: $- \frac{b}{2 a} = 4$ --> $b = - 8 a$ (1)
y-coordinate of vertex f(4) = -1
f(4) = 16a + 4b + c = - 1 (2)
The parabola passes at point (-2, 35), then f(-2) = 35
f(-2) = 4a + 2b + c = 35 (3)
We have 3 equations to find 3 unknowns a, b, and c.

(3) gives: $4 a + 2 \left(- 8 a\right) + c = - 12 a + c = 35$
->$c = 35 + 12 a$
(2) gives: $16 a + \left(- 32 a\right) + \left(35 + 12 a\right) = - 1$
-4a = - 36. --> $a = 9$
$b = - 8 a = - 72$
$c = 35 + 12 a = 35 + 108 = 143$

Equation: $y = 9 {x}^{2} - 72 x + 143 = 0$

Check: x of vertex --> x = -b/2a = 72/18 = 4 OK
y of vertex: f(4) = 9(16) - 72(4) + 142 = 144 - 288 + 143 = - 1 OK