SOLUTION: Identify the graph using co-ordinate geometry. Where are the vertex and focus located? What is the equation of the directory? 〖(y+2)〗^2=8(x-1)

Question 1200276: Identify the graph using co-ordinate geometry. Where are the vertex and focus located? What is the equation of the directory?
〖(y+2)〗^2=8(x-1)

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

"co-ordinate" should be "coordinate".
"directory" should be "directrix".

For parabolas that open left or right, we have this template
(y-k)^2 = 4p(x-h)
If p > 0, then the parabola opens to the right.
If p < 0, then the parabola opens to the left.

Compare
(y-k)^2 = 4p(x-h)
with
(y+2)^2 = 8(x-1)
to find these values
p = 2
h = 1
k = -2

Since p > 0, this particular parabola opens to the right.

The vertex is located at (h,k) = (1,-2)
This is the left-most point of the parabola. See the graph below.

The value of p is the focal distance.
It's the distance from the vertex to the focus.
It's also the distance from the vertex to the directrix.

Start at the vertex (1,-2) and move p = 2 units to the right along the axis of symmetry to arrive at the focus (3,-2).
The focus is always located inside the parabolic bowl shape, and it's also always on the axis of symmetry.

The directrix is on the opposite side of the vertex.
We start at (1,-2) and move p = 2 units left to arrive at (-1,-2).
Then draw a vertical line through (-1,-2) to represent the directrix.
The equation of the directrix is the vertical line x = -1.
The directrix is perpendicular to the axis of symmetry which has the equation y = -2.

Graph:

Vertex = (1,-2)
Focus = (3,-2)
Directrix Equation: x = -1

You can use GeoGebra to confirm those answers.

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