Need help with this!!!

Question

Consider the function         .

What are the coordinates of the vertex of this function?     

What are the coordinates of the focus of this function?     

What is the equation for the directrix of this function?   

Answer 1

The general form of a parabola:

Opening left or right: y² = 4ax (Opening left: -4ax)

Opening up or down: x² = 4ay (Opening down: -4ay)

To remember which is which: If it opens up or down, then there are 2 x values for every y value--similar logic for right and left.

If the vertex is not at the origin, then we apply offsets: If the origin is at (j, k), then replace x with x - k, and replace y with y = k

a is the distance from the vertex to either the focus or directrix. (2a is the distance from focus to directrix)

So...the steps:

1. Put the equation into the form described above
2. Find a and use that to determine the relative location of focus, vertex, and directrix
3. PLOT the function to verify your results

For this problem:

y = -1/16(x - 3)² + 2

Rearrange to get it into the general form:

-1/16(x - 3)² = y - 2

(x - 3)² = -16* (y - 2)

From this equation:

Vertex is at (3, 2)

a = -4 ---> therefore it opens down

directrix at y = 2 + 4 = 6 (equation is y = 6)

focus at y = 2 - 4 = -2

(x is the same for focus, and vertex)

I confirmed these answers by plotting in desmos (an online graphing tool)