Question

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

Answer 1

Find the equation of the parabola

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

Explanation:

Equation of the parabola : `y = ax^2 + bx + c`. Find a, b, and c
Vertex (4, -1)
x-coordinate of vertex: `-b/(2a) = 4` --> `b = - 8a` (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: `4a + 2(-8a) + c = - 12a + c = 35`
->`c = 35 + 12a`
(2) gives: `16a + (- 32a) + (35 + 12a) = - 1`
-4a = - 36. --> `a = 9`
`b = - 8a = - 72`
`c = 35 + 12a = 35 + 108 = 143`

Equation: `y = 9x^2 - 72x + 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