**In class**we worked on converting functions into rational form.

**Assignment: 9.8 on p. 555: 1-8, 11-12**

**Here is an example problem, similar to the one worked in class.**

Let's analyze the function given by the equation below. We will try to work out its various characteristics and sketch a graph before resorting to computer graphing.

To obtain a common denominator we multiply the first term by and the second term by

Now we can simplify the expressions in the numerators and re-write as a single fraction.

Which simplifies to

or

**Now that the function is in rational form, we can analyze its properties.**

**in the function. Because there are no factors common to both the numerator and denominator. i.e. . These common factors would work out to essentially multiplying the function by 1 for all values of $x$ except for when $x + 2 = 0$ at $x = -2$. For that single value $(x + 2)/(x + 2)$ becomes undefined and we get a hole in the graph. Anyway, since this is not the case, our function has**

*there are no holes**no holes*.

The function has vertical asymptotes at

and

We can tell from the numerator that the function will have x-intercepts where the numerator evaluates to zero, so we solve to get

=

So, we note the two x-intercepts.

In addition, we note that this function has denominator of greater degree than the numerator, so as approaches the denominator will overpower the numerator and the value of the function will approach , creating horizontal asymptotes at

Through point testing and some more intuition we can identify the location of the 4 sections of this function and sketch a graph