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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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