# Parabola Calculator

This calculator will find either the equation of the parabola from the given parameters or the axis of symmetry, eccentricity, latus rectum, length of the latus rectum, focus, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the entered parabola.

To graph a parabola, visit the parabola grapher (choose the "Implicit" option).

Choose what to enter:

Enter the first point:
P_1=( )
Enter the second point:
P_2=( )
Enter the third point:
P_3=( )
Choose the axis:

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

## Solution

Your input: find the equation, focus, axis of symmetry, eccentricity, latus rectum, length of the latus rectum, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the parabola that passes through the points $\left(1, 4\right)$, $\left(2, 9\right)$, $\left(-1, 6\right)$.

Assume that the equation of the parabola is $y=a x^{2} + b x + c$.

Since the parabola passes through the point $\left(1, 4\right)$, then $4=a + b + c$.

Since the parabola passes through the point $\left(2, 9\right)$, then $9=4 a + 2 b + c$.

Since the parabola passes through the point $\left(-1, 6\right)$, then $6=a - b + c$.

Thus, we have obtained the following system: \begin{cases}a + b + c=4\\4 a + 2 b + c=9\\a - b + c=6\end{cases}

Solving it (for steps, see system of linear equations calculator), we get that $a=2$, $b=-1$, $c=3$.

Thus, the equation of the parabola is $y=2 x^{2} - x + 3$.

Equation of the parabola: $y=2 x^{2} - x + 3$.

Vertex form: $y=2 \left(x - \frac{1}{4}\right)^{2} + \frac{23}{8}$.

No intercept form.

Vertex: $\left(\frac{1}{4},\frac{23}{8}\right)$.

Focus: $\left(\frac{1}{4},3\right)$.

Eccentricity: $1$.

Directrix: $y=\frac{11}{4}$.

Latus rectum: $y=3$.

The length of the latus rectum: $\frac{1}{2}$.

Axis of symmetry: $x=\frac{1}{4}$.

Focal parameter: $\frac{1}{4}$.

No x-intercepts.

y-intercept: $\left(0, 3\right)$.

Graph: to graph the parabola, visit the parabola graphing calculator (choose the "Implicit" option).