If my focus is given as (0, -3/2), how do I use that to find the standard equation of the parabola?

Question

I know the answer is x^2=-6y (or that is the given answer), but I need to know how they got it. Thanks.

PS, the vertex is 0,0. My professor didn't go over this at all and if I see one worked out I'm sure I can do the rest.

I'm asking to see how this problem is worked so I can know how to do the rest, so please don't post links. Thanks in advance.

To the first answerer, how do you derive the answer? What formula or equation do you use?

Answer 1

For the parabol use the following equations...

standard equation parabol : 4py = x^2
F : (0,p)
direction line : y=-p

Answer 2

The directed distance between the focus and the vertex of a parabola is given by 1/(4a). [The distance between your vertex (0, 0) and focus (0, -3/2) is 3/2. The focus is located directly below your vertex.]

Your parabola is a downward-opening one, with vertex (0, 0). This means your equation, y = a(x - h)² + k, will have (h, k) = (0, 0).

1/(4a) = -3/2. Solve for a.
-12a = 2,
a = -1/6

This makes your equation:
y = (-1/6)(x - 0)² + 0, or
y = (-1/6)x².

Answer 3

Louise has it

Just look at the general equation for a quadratic since this is what produces a parabolic graph:

f(x)=ax^2+bx+c, a≠0
but there is another form that is easier to understand when determining more information of the graph:

f(x) = a(x - h)^2 + k; a, h, k constants, a≠0

note that a(x-h)^2 is never negative so a determines the direction of the opening.

but parabolas have specific properties and we have

Axis of symetry: x = -b/(2a)

Vertex: [ -b/(2a), f(-b/(2a))]

Max or Min: f(-b/(2a)), max if a > 0, min if a < 0

y-intercepts: f(0) = c
x-intercepts: where f(x) = 0, (if they exist)

Answer 4

here the x coordinate is 0 therefore the general eqn of the parabola is

x^2 = 4 a y -------------- (i) (where 4a is the length of latus rectum)

the focus is (0 , a)
therefore here a = -3/2

putting this value of a in general eqn (i)

we get

x^2= 4 * (-3/2) * y

=>x^2 = -6y Ans.

Answer 5

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Answer 6

-6=x

think if u do -6x-6 u get 36
and if u do -6^2 u get 36

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