Equations of Simple Parabolas

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EQUATIONS OF SIMPLE PARABOLAS

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If a parabola is placed in a coordinate plane in a simple way,
then a simple equation is obtained, as derived below.

Place a parabola with its directrix parallel to the x-axis,
its focus on the y-axis,
and its vertex at the origin, as shown at right.

Let p denote the y-value of the focus.
Let (x,y) be a typical point on the parabola.

The distance from (x,y) to the focus is found using the distance formula, and is:
(x-0) 2 + (y-p) 2 = x 2 + (y-p) 2
The distance from (x,y) to the directrix is: (x-x) 2 + (y-(-p)) 2 = (y+p) 2
These two distances must be equal:
x 2 + (y-p) 2 = (y+p) 2

Squaring:	x 2 + (y-p) 2= (y+p) 2
Multiplying out:	x2 + y2 -2py+ p2 = y2 + 2py+ p2
Subtracting y2 +p2 :	x2 -2py = 2py
Adding 2py :	x2 = 4py
Dividing by 4p and rearranging:	y= 1 4p x2

CONCLUSIONS

Every equation of the form y= ax2 (for a≠0) is a parabola with directrix parallel to the x-axis, focus on the y-axis, and vertex at the origin.

If a>0 , the parabola is concave up (holds water).
If a<0 , the parabola is concave down (sheds water).

Setting a= 1 4p and solving for p shows that the coordinates of the focus are (0,1 4a) .

SHIFTING THE PARABOLA

Shift the parabola (together with its focus and directrix) horizontally by h , and vertically by k .
This yields the following information:

original equation:	y=a x2	shifted equation:	y=a (x-h) 2 +k
original vertex:	(0,0)	new vertex:	(h,k)
original focus:	(0,p) =(0,1 4a)	new focus:	(h,p+ k)=(h,1 4a+ k)
original directrix:	y=-p	new directrix:	y=-p+ k

EXAMPLE (graphing a shifted parabola)

Consider the equation y=-3 (x+5) 2 +1 .
Since a=-3< 0 , the parabola sheds water (is concave down).
The vertex is (-5,1 ) .
The distance from the focus to the vertex is |p| = |14 a| = | 14(-3)| = 112 .
Thus, the focus is (-5,1- 112) =(-5, 1112) and the equation of the directrix is y=1+1 12 , i.e., y=13 12 .
Plotting an additional (easy) point gives information about the width of the parabola:
when x=0 , y=-3 (0+5) 2 +1=-74 .
In this parabola, the vertex is close to the focus, so the parabola is narrow.

y=-3 (x+5) 2 +1
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