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Parabola Calculator
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A parabola is a set of all points in the plane, which are equidistant from a fixed line and a fixed point not on the line. The line is called directrix of the parabola, and the point is called the focus. Parabola Calculator is an online tool which makes calculations easy and fast. Try our free Parabola Calculator, understand the various steps involved in solving problems and work on examples based on the concept you need to understand.

Below is given a default equation, click "Submit". The calculator finds the vertex, focus and directrix of the parabola by its appropriate formulas.

## Steps for Parabola

Step 1 :

Observe the given equation of parabola.

Step 2 :

If the equation of parabola is in standard form(y = ax2 + bx + c), then apply the formula:
focus = ( $\frac{-b}{2a}$ , c - $\frac{(b^{2} - 1)}{4a}$)
vertex = ($\frac{-b}{2a}$, $\frac{-D}{4a}$). Where D = b2 - 4ac
parabola may also be characterized as a conic section with an eccentricity of 1
directrix => y = c -  $\frac{(b^{2} + 1)}{4a}$

If the equation of parabola is in vertex form(y = a (x -h)2 + k), then subtitute a = $\frac{1}{4p}$, b = $\frac{-h}{2p}$ and c = $\frac{h^{2}}{4p}$ + k
and apply the formula:
focus = ( $\frac{-b}{2a}$ , c - $\frac{(b^{2} - 1)}{4a}$).
vertex = ($\frac{-b}{2a}$, $\frac{-D}{4a}$). Where D = b2 - 4ac
parabola may also be characterized as a conic section with an eccentricity of 1
directrix => y = c -  $\frac{(b^{2} + 1)}{4a}$

Step 3 :

Plug in the values in the suitable formula and solve further.

## Examples on Parabola Calculator

1. ### Find the vertex of the parabola : y = x2 - 2x + 1?

Step 1 :

Given equation is in standard form(y = ax2 + bx + c): y = x2 - 2x + 1
Where a = 1, b = -2 and c =1

Step 2 :

vertex = ($\frac{-b}{2a}$, $\frac{-D}{4a}$). Where D = b2 - 4ac

D = b2 - 4ac
D = (-2)2 - 4 (1)(1)
D = 4 - 4
D = 0

Step 3 :

So, vertex = ($\frac{-b}{2a}$, $\frac{-D}{4a}$)
vertex = ($\frac{-(-2)}{2(1)}$, $\frac{0}{4(1)}$)
vertex = (1, 0)

vertex = (1, 0)

2. ### Find the directrix of the parabola : y = 3(x + 3)2 + 4?

Step 1 :

Given equation is in vertex form(y = a (x -h)2 + k) =>  y = 3(x + 3)2 + 4
where a = 3, h = -3 and k = 4

Step 2 :

a = $\frac{1}{4p}$ => 3 = $\frac{1}{4p}$ => p = $\frac{1}{12}$
b = $\frac{-h}{2p}$ => b = $\frac{3 \times 12}{2}$ = $\frac{36}{2}$ = 18
c = $\frac{h^{2}}{4p}$ + k => $\frac{9 \times 12}{4}$ + 4 => 27 + 4 = 31

Step 3 :

focus = ( $\frac{-b}{2a}$ , c - $\frac{(b^{2} - 1)}{4a}$)
focus = ( $\frac{-18}{2(3)}$ , 31 - $\frac{(18^{2} - 1)}{4(3)}$)
focus = (-3, 4.083)