A parabola contains the points (-3,-18), (0,3), and (3,6). What is the equation of this parabola in standard form?

Question

ive been stuck in this question for the past hour, I really need help

Answer 1

2 possible standard parabolas, through those points. one opening downward with (3,6) as the vertex and maximum point and the other opening rightward with (-3,-18) as the vertex with x=-3 the minimum x coordinate.

for the 1st, the general equation is y-h = a(x-k) with (k,h) = vertex = (3,6), so

y-6 = a(x-3) with a<0 or negative

plus either other point in to solve for a

3-6 = a(6-3)^2

-3 = 9a

a =-1/3

the equation is

y-6=-(1/3)x-3)^2

for the 2nd the general equation is x-k = a(y-h)^2

do the same method

plug in the vertex (-3,-18) to get

x+3 = a(y+18)^2

plug in either of the other 2 points to solve for a

6+3 = a(3+18)^2

9 = 441a

a = 9/441 = 1/49

x+3 = (1/49)(y+18)^2

there are an infinite number of other parabolas through those 3 points, but at angles other than vertical or horizontal

they will all have xy terms in the equation