Question

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

Answer 1

Solve for the asymptotic discontinuities

Explanation:

To make it easier to look for the discontinuities, it helps to completely factor all the expressions first.
`(-2x^2+1)/(2x^3+4x^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.
`2x^2(x+2)=0`

We can separate this into two equations:

Equation 1:
`2x^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.