How do you find the vertex, focus, and directrix of a parabola (y-9)^2 = -8(x+5)?

Apr 17, 2015

A Guide
If you got a parabola in the form ${\left(y - {y}_{v}\right)}^{2} = 4 a \left(x - {x}_{v}\right)$

Then $\left({x}_{v} , {y}_{v}\right)$ is your vertex,

Focus is $\left({x}_{v} + a , {y}_{v}\right)$

and directrix is the line $x = {x}_{v} - a$

Solution to problem

${\left(y - 9\right)}^{2} = - 8 \left(x + 5\right)$

$\equiv {\left(y - 9\right)}^{2} = 4 \left(- 2\right) \left(x - \left(- 5\right)\right)$

Vertex is $\left(- 5 , 9\right)$

It can be seen by comparison with the general form that $a = - 2$

Implying that,

Focus is $\left(- 5 + \left(- 2\right) , 9\right)$ or $\left(- 7 , 9\right)$

Directrix is $x = - 5 - \left(- 2\right)$ or $x = - 3$

Verify on the graph
graph{sqrt(-8x-40)+9 [-6.75, 13.25, -4.24, 5.76]}