The free online Parabola Calculator displays the graph for the given parabola equation. The Mathematics Master online parabola calculator tool speeds up the calculation and shows the parabola graph in seconds. This parabola equation calculator speeds up and simplifies your calculations by solving the parabolic equation's related properties.

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

How to graph a Parabola?

You can use an online parabola graph calculator to plot the graphical representation of the given equation for quick and easy calculations. However, some steps must be taken to plot a parabola graph manually:

  • First, locate the x and y intercepts.
  • Look for extra points so you can plot a graph with at least five points.
  • Plot the points and draw your parabola graph.

What is a Latus Rectum?

Latus Rectum is the focal chord that runs perpendicular to the parabola’s axis and through the focus. LL’ = 4a is the formula for calculating the length of the latus rectum. The latus rectum endpoints are represented by (a, 2a) (a, -2a).

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