Parabola Calculator

What is a Parabola?

A parabola is a U-shaped curve representing the quadratic function \( f(x) = ax^2 + bx + c \). When the value of a is less than zero, the graph of the parabola is downward (or opens down). When the value of a is greater than zero, the parabola graph goes up (or opens up). All points in the parabola are equidistant from the directrix and the focus.

How to find the Equation of a Parabola?

The equation of a parabola in vertex form can be found in two steps:

Step 1: Using the vertex’s (known) coordinates, (h, k), write the parabola’s equation in the form: \( y = a(x − h)^2 + k \)

Step 2: Determine the coefficient of a value by inputting point P coordinates into the equation from step 1 and solving for a.

Example

Find the points of intersection of the two parabolas with equation \( y = -(x – 2)^2 + 2 \) and \( y = x^2 – 3x + 1 \).

The points of intersection of the two parabolas are solutions to the simultaneous equations \( y = -(x – 2)^2 + 2 \) and \( y = x^2 – 3x + 1 \).

\( -(x – 2)^2 + 2 = x^2 – 3x + 1 \)

\( -2x^2 + 7x – 3 = 0 \)

\( -2x^2 + 6x + x – 3 = 0 \)

Solutions: x = 3 and x = 0.5

Use one of the equations to find y:

x = 3 in the equation \( y = -(x – 2)^2 + 2 \) to obtain \( y = – (3 – 2 )^2 + 2 = 1 \)

x = 0.5 in the equation \( y = -(x – 2)^2 + 2 \) to obtain \(y = – (0.5 – 2)^2 + 2 = – 0.25\)

Points: (3 , 1) and (0.5 , – 0.25)

FAQs