This calculator finds the equation of parabola with vertical axis given its vertex of the parabola and a point on the parabola.
Formulas Used in the Calculator
The equation of a parabola whose vertex is given by its coordinates \( (h,k) \) is written as follows
\[ y = a(x - h)^2 + k \]
For the point with coordinates \( A = (x_0 , y_0) \) to be on the parabola, the equation \( y_0 = a (x_0 - h)^2 + k \) must be satified.
Solve the above equation to find coefficient \( a \)
\[ a = \dfrac{y_0 - k}{(x_0 - h)^2} \]
Note that
1) if \( h = x_0 \), the denominator in \( a \) is equal to zero and the problem has no solution because both the vertex and the given point \( A \) are in the same vertical line.
2) if \( k = y_0 \), there is no parabola because both the vertex and the given \( A \) are in the same horizontal line.
How to Use the Calculator?
1 - Enter the \( h \) and \( k\) coordinates the vertex and the coordinates \( x_0 \) and \( y_0 \) of the point on the parabola and press "Calculate".
Three equations are displayed: in vertex form as given above, an exact one (middle one) where the coefficients are in fractional forms a third equation with approximated (if necessary) coefficients in decimal form.
You may also change the number of decimal places.
The problem has no solution if \( h = x_0 \) or \( k = y_0 \)