SOLUTION: Find the vertex, focus, directrix, and axis of symmetry of each parabola. x^2+8y+4x-4=0 We have been using the formulas y-k=a(x-h)^2 and x-h=a(y-k)^2 I don't understand which to

Question 193521This question is from textbook algebra and trigonometry structure and method book 2
: Find the vertex, focus, directrix, and axis of symmetry of each parabola.
x^2+8y+4x-4=0
We have been using the formulas y-k=a(x-h)^2 and x-h=a(y-k)^2 I don't understand which to use here, even after completing the square and its confusing because I don't know which is which. please explain and be specific. thank you. This question is from textbook algebra and trigonometry structure and method book 2

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Rearrange the terms.

Now we need to complete the square for the "x" terms.

Take half of the x coefficient 4 to get 2. Square 2 to get 4.

Add the previous result 4 to both sides.

Subtract 4 from both sides.

Combine like terms.

Factor to get

Subtract from both sides

Factor out the GCF 8 from the left side

Multiply both sides by

Rewrite as

Now the equation is in the form where , and

Vertex:

The vertex of is (h,k). Since and , the vertex of is (-2,1)

-----------------------------------------------------------------------

Focus:

First, we define where the absolute value of "p" is the distance from the vertex to the focus.

Note: the formula is derived from the equation

The focus of is the point

. In other words, simply add the value of "p" to the y coordinate of the vertex to get the focus.

So let's find "p": . So

Since , , and , this means that

So the focus of the form

is the point (-2,-1)

------------------------------------------------------------

Directrix:

The directrix of is the equation .

Since , this means that the directrix is

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Axis of Symmetry:

In this case, the axis of symmetry is simply the equation of the vertical line through the vertex.

Since , this means that the axis of symmetry is

Visual Check:

Graph of with the vertex, focus, directrix (green) and axis of symmetry (blue)

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