How do you find focus, directrix and axis of symmetry?

Question

I have a test on chapter 9 in Algebra 2 tomorrow and I don't understand how to find the focus, directrix, and axis of symmetry of a equation. For example:

 

-5+1/3y²=0

Answer 1

Hi Emmy, 

 

To find the focus, directrix, and axis of symmetry of a parabola, first find the vertex of the parabola by completing the square. 

 

By the way, your example is is missing an x-variable, so I will give a simple example to show the process. 

 

Let's find the key features of 2y²-2x+2y=4. Perform the following: 

 

1) Isolate each term so that the squared variable is on one side [Note: Since the y is squared here, this equation has a vertical axis of symmetry, meaning the graph opens up or down].

 

We get 2y²+2y=2x+4 --> 2(y²+y)=2x+4 --> y²+y=x+2

 

2) Complete the square

 

We get (y+1)²=x+3 (because (y+1)²=y²+y+1 and we need to add one to both sides since we originally only had y²+y on the left side)

 

The standard form of a parabola with a vertical axis of symmetry [like my example] is (y-k)²=4p(x-h), where p is the directed distance between the vertex and focus (and also between the vertex and directrix).   

 

For vertical axes of symmetry:

The vertex is given by V=(h,k)                    

The focus is given by F=(h, k+p)

The directrix is given by y=k-p

The axis of symmetry is given by x=h

 

Here, k must equal -1 to give us y+1 in our square, and h must equal -3 to give us x+3 on the right side. Thus we get:

Vertex: (-3, -1)

Since we have a coefficient of 1 in front of x, we have 4p=1--> p=1/4 and  

Focus: (-3, -3/4)

Directrix: y=-5/4

Axis of symmetry: x=-3

 

 

For a horizontal axis of symmetry (so the graph opens to the left or right), we have the following: 

 

The standard form of a parabola with a horizontal axis of symmetry (y-k)2=4p(x-h), where p is the directed distance between the vertex and focus (and also between the vertex and directrix).

For horizontal axes of symmetry:

The vertex is given by V=(h,k)
The focus is given by F=(h+p, k)
The directrix is given by x=h-p
The axis of symmetry is given by y=k

 

You would again find p and use the same process like above.

 

I know that was a bit long, but I this explains every key feature of parabolas. Good luck with everything!

 

-Matt

Answer 2

Parabola: y = ax^2 + bx + c

 

Axis of symmetry: x = -b/2a

 

Focus: a point that is one unit away from the turning point of the parabola & it's located inside of the parabola

 

Directrix: a line that is orthogonal to the axis of symmetry & has a point one unit away from the turning point.

 

Example: y = x^2 - 8x +12

 

Axis of symmetry: x = 8/2 = 4

 

Turning point (4, -4)

 

Focus: (4, -3)

 

Directrix: y = -5