# Focus of a Parabola

A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the **focus** of the parabola and the line is called the directrix.

The focus lies on the axis of symmetry of the parabola.

### Finding the focus of a parabola given its equation

If you have the equation of a parabola in vertex form $y=a{\left(x-h\right)}^{2}+k$, then the vertex is at $\left(h,k\right)$ and the focus is $\left(h,k+\frac{1}{4a}\right)$.

Notice that here we are working with a parabola with a vertical axis of symmetry, so the $x$-coordinate of the focus is the same as the $x$-coordinate of the vertex.

**Example 1:**

Find the focus of the parabola $y=\frac{1}{8}{x}^{2}$.

Here $h=0$ and $k=0$, so the vertex is at the origin. The coordinates of the focus are $\left(h,k+\frac{1}{4a}\right)$ or $\left(0,0+\frac{1}{4a}\right)$.

Since $a=\frac{1}{8}$, we have

$\frac{1}{4a}=\frac{1}{\left(\frac{1}{2}\right)}$

$=2$

The focus is at $\left(0,2\right)$.

**Example 2:**

Find the focus of the parabola $y=-{\left(x-3\right)}^{2}-2$.

Here $h=3$ and $k=-2$, so the vertex is at $\left(3,-2\right)$. The coordinates of the focus are $\left(h,k+\frac{1}{4a}\right)$ or $\left(3,-2+\frac{1}{4a}\right)$.

Here $a=-1$, so

$-2+\frac{1}{4a}=-2-\frac{1}{4}$

$=-2.25$

The focus is at $\left(3,-2.25\right)$.

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