# How do you graph and find the discontinuities of F(x) = (-2x^2 + 1)/(2x^3 + 4x^2)?

Sep 21, 2015

Solve for the asymptotic discontinuities

#### Explanation:

To make it easier to look for the discontinuities, it helps to completely factor all the expressions first.
$\frac{- 2 {x}^{2} + 1}{2 {x}^{3} + 4 {x}^{2}}$
=(-2x^2+1)/(2x^2(x+2)

The discontinuities of a rational function are asymptotic discontinuities. These are the values of of $x$ that will cause the denominator to be 0 (making it undefined). To solve for the asymptotic discontinuities, equate the denominator to 0 and find the solutions.
$2 {x}^{2} \left(x + 2\right) = 0$

We can separate this into two equations:

Equation 1:
$2 {x}^{2} = 0$
$x = 0$

Equation 2:
$x + 2 = 0$
$x + 2 - 2 = 0 - 2$
$x = - 2$

The asymptotes are $x = 0$ and $x = - 2$. Graph these two lines on your paper using a dotted line.

As for graphing, you can do that by creating a table of values.

It should end up like this:
graph{(-2x^2+1)/(2x^3+4x^2) [-10, 10, -5, 5]}

You will notice that the graph looks like it's approaching x=0 and x=-2, but it will never actually touch it.