# How do you identity if the equation y^2+18y-2x=-84 is a parabola, circle, ellipse, or hyperbola and how do you graph it?

Apr 1, 2017

It is a parabola.

#### Explanation:

An equation which can be written in the form of $y = a {x}^{2} + b x + c$ or $x = p {y}^{2} + q y + r$ is a parabola and to draw it you have to just convert it into vertex form such as $y = a {\left(x - h\right)}^{2} + k$ or $x = p {\left(y - k\right)}^{2} + h$. Here $\left(h , k\right)$ is the vertex.

Here we have ${y}^{2} + 18 y - 2 x = - 84$ or $2 x = {y}^{2} + 18 y + 84$ and in vertex form it is

$x = \frac{1}{2} \left({y}^{2} + 18 y\right) + 42$

or $x = \frac{1}{2} \left({y}^{2} + 2 \times 9 y + {9}^{2} - {9}^{2}\right) + 42$

or $x = \frac{1}{2} {\left(y + 9\right)}^{2} - \frac{81}{2} + 42$

or $x = \frac{1}{2} {\left(y - \left(- 9\right)\right)}^{2} + \frac{3}{2}$

Here $y + 9 = 0$ is the axis of symmetry and as coefficient of ${\left(y + 9\right)}^{2}$ is $\frac{1}{2}$, a positive number, it opens up towards right. Its vertex is $\left(\frac{3}{2} , - 9\right)$.

We can now choose some values of $y$ around $- 9$, say $\left\{- 6 , - 7 , - 8 , - 10 , - 11 , - 12\right\}$ and find corresponding $x$ which are $\left\{6 , \frac{7}{2} , 2 , 2 , \frac{7}{2} , 6\right\}$.

This gives us points $\left(6 , - 6\right)$, $\left(\frac{7}{2} , - 7\right)$, $\left(2 , - 8\right)$, $\left(\frac{3}{2} , - 9\right)$, $\left(2 , - 10\right)$, $\left(\frac{7}{2} , - 11\right)$, $\left(6 , - 12\right)$ and joining them gives us the following graph.

graph{y^2+18y-2x=-84 [-10.5, 29.5, -18.64, 1.36]}