# How to Write the Equation of Parabola?

In this article, you will learn how to write the equation of a parabola in standard form. 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.

## 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: $$k+p=6, k=4 → 4+p=6 →p=2$$

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

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