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
Now that the function is in rational form, we can analyze its properties.
The function has vertical asymptotes at
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