###### Example 13.6.1 Graphs of Rational Functions

In an apocalypse, a zombie infestation begins with \(1\) zombie and spreads rapidly. The population of zombies can be modeled by \(Z(x)=\frac{200000x+100}{5x+100}\text{,}\) where \(x\) is the number of days after the apocalypse began. Use technology to graph the function and answer these questions:

How many zombies are there \(2\) days after the apocalypse began?

After how many days will the zombie population be \(20{,}000\text{?}\)

As time goes on, the population will level off at about how many zombies?

We will graph the function with technology. After adjusting window settings, we have:

To find the number of zombies after \(2\) days, we locate the point \((2,3637.27)\text{.}\) Since we can only have a whole number of zombies, we round to \(3{,}637\) zombies.

To find the number of days it will take for the zombie population reach \(20{,}000\text{,}\) we locate the point \((19.999,20000)\) so it will take about 20 days.

When we look far to the right on the graph using technology we can see that the population will level off at about \(40{,}000\) zombies.