10.2 - The Parabola

Amanda T.

ICM Honors, 5

Well first of all... 

Standard Formula when the directrix is parallel to the y-axis:  (y-k)² = 4p(x-h) , and when the directrix is parallel to the x-axis, the standard formula is:  (x-k)² = 4p(y-h)

•  (h,k) is the vertex

•  p is the distance from the vertex to the focus                    

General form of the equation of a parabola is y²+Dx+Ey+F=0 when the directrix is parallel to the y-axis, or x²+Dx+Ey+f=0.

So, let's try one..

Krista's boyfriend gave her a beautiful red rose when he came to pick her up for the night.  As she was looking at the rose, thinking about how sweet he was for bringing it, she noticed what a perfect parabola the rose made (the vertex at the stem).  Quickly, in her head, she figured out that the equation for the parabola formed was (x-2)²=8(y+1).  Find the vertex, focus, and the directrix of the equation if it were graphed.  

To Solve:

To find the vertex, look at what is INSIDE the parentheses.  You'll be able to find the coordinate from that.  From (x-2), the -2 means that the x coordinate will be 2.  Always take the opposite number from what the equation gives.  Given that information, the y coordinate will be -1.  

VERTEX: (2, -1)

Since X is squared, the directrix of the parabola will be parallel to the x-axis.  Look at the standard form equation again.. (x-h)²=4p(y-k) .. find p.

8=4p

p=2

Since p is positive, the parabola opens upward.  And since p=2, the distance from the vertex to the focus is 2. So what's the coordinate of the focus then?

FOCUS: (2, 1)

The distance between the vertex and the focus is the same as the distance between the vertex and the directrix, so to find the equation for the directrix, subtract 2 from the y coordinate of the vertex.  And that would make the equation of the directrix what?

DIRECTRIX: y=-3

Now, graph it!

A mistake is evidence that somebody tried to accomplish something.

~Anonymous~

This quote helps us remember that even though we make mistakes, at least we are trying.

For more help with parabolas, you might like to visit: school.discovery.com/homeworkhelp/ webmath/parabolas.html

BACK