Question

Use the graph to answer the questions.

1) write into vertex form.

2) identify the vertex, focus, and directrix.

Answer 1

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

Vertex (-5,-4)

directrix y = -33/8

focus (-5, -31/8)

Explanation:

First identify the vertex, which is the minimum

Vertex = (-5,-4)

The vertex form is y = a( x-h)^2 + k

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

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

We need to determine a

Substitute a point on the graph

(-3,4) is on the graph

4 = a( -3+5)^2 -4

4 = a( 2)^2 -4

4 = 4a -4

Add 4 to each side

8 = 4a

Divide by 4

8/4 = a

a=2

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

To find the focus and directrix, write in standard form

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

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

1/2 (y+4) = (x+5)^2

4p = 1/2

p = 1/8

The focal length is 1/8

Subtract this from the y coordinate to get the directrix

y = -4 -1/8

y = -32/8 - 1/8 = -33/8

Add this to the y coordinate to get the coordinate for the focus the x coordinate the same

(-5, -4+1/8)

(-5, -32/4+1/8)

(-5, -31/8)