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.

PLEASE HELP ME!!!!!!!!

THANK YOU!!!!

KennedyPape
Apr 18, 2018

#1**+1 **

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

CPhill
Apr 19, 2018