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Okay so the question is...

Write the equation of a parabola with focus  and directrix . Show your work, including a sketch.

I always look up the answer before I do a question (Not the steps) so I can work through the problem and learn it the right way the first time instead of thinking i have the answer and its completly wrong and then I dont understand what went wrong (sorry for my ramble). Anyway the answer is supposed to be y=x^2/4 + x + 4. (I also checked this answer multiple times to see if i had put the equation in wrong and even went to different websites because Im so stumbled with this question.)

This is my answer so far...

The formula to find the equation of the parabola using focus and directrix. The vertex being (h, k) and (p) the distance between the vertex and the focus.

4p(y-k)=(x-h)^2

Vertex being (-2, 3).

(-2, [y coordinate of the focus + y value of the directrix] /2) = (-2, [4+2] /2) = (-2, 6/2) = (-2, 3)

(p) is the distance between the focus (-2, 4) and the vertex (-2, 3) which is 1.

p=1

Plug in the values.

4(1)(y-3) = (x-(-2))^2 = 4(y-3) = (x+2)^2

So this is were i have been getting stuck... I cant seem to simplfy/solve 4(y-3) = (x+2)^2 to y=x^2/4 + x + 4

I keep ending up with equations like y=(x^2 + 16)/4 which then equals y=(x^2)/4 + 4 and when i check that as my answer it gives me the focus of (0, 5) and directrix y=3, which isn't right.

THANK YOU!!!!

Apr 18, 2018
edited by KennedyPape  Apr 19, 2018

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Let's work backwards

y  = x^2/4  +  x + 4     subtract 4 from both sides

y - 4  =  x^2/4 + x      factor out the 1/4 on the right side

y - 4   = (1/4)(x^2 + 4x)    complete the square on x within the parentheses

y - 4  = (1/4)(x^2 + 4x  + 4  - 4)   factor the first three terms inside the parentheses

y - 4  = (1/4) [ (x + 2)^2 - 4 ] distibute the 1/4 across the brackets

y - 4  = (1/4)(x + 2)^2  -1     add 1  both sides

(y - 3)  = (1/4) (x + 2)^2

In the form (y - k) = a(x - h)^2....the vertex is at  ( -2, 3)

Multiply both sides by 4

4(y - 5) = (x + 2)^2

In the form

4p (y - k)  = (x - h)^2

p  = the focal distance from the vertex

So...in our equation    4  = 4p   ⇒  p  = 1

Since the parabola turns upward....the focus is given by (h, k + p)  = (-2,  3 + 1) =

(-2, 4)

The directrix is given by   y  = k - p   =  3 - 1  =  2

Look at the graph here : https://www.desmos.com/calculator/wixfeughs1

Both functions are exactly the same

The vertex  is  (-2, 3)

The focus is (-2,4)

The directrix  is  y  = 2

Apr 19, 2018
edited by CPhill  Apr 19, 2018