How to Write the Equation of Parabola?

A parabola is a U-shaped line or curve that is defined as a locus of points that the distance to a fixed point (the focus) and a fixed straight line (the directrix) are equal.

Related Topics

• How to Find the Focus, Vertex, and Directrix of a Parabola

Step by Step Guide to Write the Equation of Parabola

• The standard form of Parabola when it opens up or down is \((x- h)^2= 4p(y-k)\), where the vertex is \(h, k\), the focus is \(h,k+p\) and the directrix is \(y=k-p\).
• The standard form of Parabola when it opens right or left is \((y-k)^2= 4p(x-h)\), where the vertex is \(h, k\), the focus is \(h+p,k\) and the directrix is \(x=h-p\).

Equation of Parabola – Example 1:

Write the equation of the parabola with vertex \((1, 4)\) and focus \((0, 6)\)

Solution:

The standard form of Parabola is \((x- h)^2= 4p(y-k)\).

Vertex is \((1, 4)\) then: \(h=1, k=4\)

Focus is \((0, 6)=(h, k+p)\), then: \(k+p=6, k=4 → 4+p=6 →p=2 \)

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Plugin the values in the equation: \((x- 1)^2= 4(2)(y-4)\)

Then: \((x- 1)^2= 8(y-4)\)

Exercises for Writing Equation of Parabola

Write the equation of each Parabola.

• Vertex \((2, 5)\) and focus \((2, 7)\)
• Vertex \((3, 0)\) and focus \((5, 0)\)
• Vertex \((-1, 2)\) and focus \((-1, 0)\)
• Vertex \((0, 3)\) and focus \((-3, 3)\)

• \(\color{blue}{(x-2)^2=8(y-5)}\)
• \(\color{blue}{y^2=8(x-3)}\)
• \(\color{blue}{(x+1)^2=-8(y-2)}\)
• \(\color{blue}{(y-3)^2=-12x}\)