Parabola Calculator Given its Vertex and a Point

Formulas Used in the Calculator

This calculator determines the equation of a vertical parabola given the coordinates of its vertex and one other point on its graph[cite: 1].

The equation of a parabola with vertex \( (h, k) \) can be written in vertex form as:

\[ y = a(x - h)^2 + k \]

For a given point \( A = (x_0, y_0) \) to lie on the parabola, its coordinates must satisfy the equation:

\[ y_0 = a(x_0 - h)^2 + k \]

Solving this equation algebraically yields the coefficient \( a \):

\[ a = \dfrac{y_0 - k}{(x_0 - h)^2} \]

Important Constraints:

• If \( h = x_0 \), the denominator for \( a \) is zero. The problem has no solution because the vertex and the given point lie on the same vertical line.
• If \( k = y_0 \), it results in \( a = 0 \). This represents a horizontal line, meaning no vertical parabola exists through these points.