Translated Parabolas

Equation Forms for Translated Parabolas

A parabola with vertex (h, k) has an equation of the following form.

$\left(x-h\right)^2=4\left(y-k\right)\:$or $\left(y-k\right)^2=4\left(x-h\right)$.

The focus is distance $\mid p\mid$ from the vertex.

Example

Write an equation for the parabola with vertex (1, 3) and focus (– 1, 3) and graph it. Give the domain and range.

Solution

Since the focus is to the left of the vertex, the axis of symmetry is horizontal and the parabola opens to the left.

The directed distance between the vertex and the focus is –1 –1, or –2, so p = –2 (since the parabola opens to the left).

$\left(y-k\right)^2=4p\left(x-h\right)$

Substitute for p, h, and k.

$\left(y-3\right)^2=4\left(-2\right)\left(x-1\right)$

Multiply.

$\left(y-3\right)^2=-8\left(x-1\right)$

The domain is $\left(-\infty,1\right]$, and the range is $(-\infty ,\infty )$.