How do you identify the vertex, focus, directrix and the length of the latus rectum and graph #4(x-2)=(y+3)^2#?

1 Answer
Oct 30, 2016

Please see the explanation.

Explanation:

Rewrite the equation in the vertex form

#x = a(y - k)^2 + h#

where #(h, k)# is the vertex and the signed distance of the vertex to the focus is #f= 1/(4a)#

Divide by both sides by 4:

#x - 2 = 1/4(x + 3)^2#

Write the + as a - -

#x - 2 = 1/4(x - -3)^2#

Add 2 to both sides:

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

Now, that the equation is in standard form, please observe that the vertex is at #(2, -3)#

The distance, in the x direction, from the vertex from to the focus is:

#f = 1/(4(1/4)) = 1#

Therefore, we add 1 to the x coordinate of the vertex and we see that the focus is #(3, -3)#

The directrix is the vertical line the same distance in the opposite direction from the vertex so we subtract 1 from the x coordinate of the vertex; making its equation:

#x = 1#

Modifying the equation, #f = 1/(4a)#, to be #a = 1/(4f)#, the length of the latus rectum is the denominator, #4f#:

#4f = 4(1) = 4#