2.6: Rational Functions and Asymptotes

Section 2.6 is an introduction to rational functions.

The definition of a rational function is: "any function which can be written as the ratio of two polynomial functions."

According to our book, a rational function can be written in the form

where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.

We learned form Chapter 1 that the domain of a rational function of x includes all real numbers except x-values that make the denominator zero.

The other topic of 2.6 is asymptotes.

The definition of a vertical asymptote is "A vertical line which the graph of the line of a function approaches but never reaches."

The definition of a horizontal asymptote is "A horizontal line which the graph of a function approaches as variable tends to positive or negative infinity. It should be noted that the graph can cross the horizontal asymptote as many times as it likes (as with many oscillating functions). A horizontal asymptote occurs when the limit of a function as the variable approaches either positive or negative infinity is a constant."

Let "f" be the rational function f(x) where

where

N(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ and

D(x) = b_mx^m + b_m-1x^m-1 + ... + b₁x + b₀

Where N(x) and D(x) have no common factors.
1. The graph of "f" has vertical asymptotes at the zeros of D(x).
2. The graph of "f" has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).
a. If n is less than m, the line y = 0 (x-axis) is a horizontal asymptote.
b. If n = m, the line is a horizontal asymptote.

c. If n is greater than m, the graph of "f" has no horizontal asymptote.

Example: Find the Domain and Asymptotes

1. Find the vertical asymptotes by taking D(x) and setting it equal to zero.

(x - 2)³ = 0
x - 2 = 0
x = 2

therefore a vertical asymptote is the line x = 2

2. Find the horizontal asymptotes
n = 0 and m = 3 so n is less than m, therefore the line y = 0 is the horizontal asymptote.

3. Graph it using a graphing utility.

4. Domain (- ∞, 2) U (2, ∞)

5. Range (- ∞, 0) U (0, ∞)

The x-intercepts are where the graph crosses the x-axis, and the y-intercepts are where the graph crosses the y-axis.

More specifically,

• an x-intercept is a point in the equation where the y-value is zero, and
• a y-intercept is a point in the equation where the x-value is zero.
  

• Find the x- and y-intercepts of 25x² + 4y² = 9

Using the definitions of the intercepts, I will proceed as follows:

x-intercept(s):

y = 0 for the x-intercept(s), so:

25x² + 4y² = 9
25x² + 4(0)² = 9
25x² + 0 = 9
x² =


x = ± ( )




Then the x-intercepts are the points (, 0) and ( , 0)

y-intercept(s):

x = 0 for the y-intercept(s), so:

25x² + 4y² = 9
25(0)² + 4y² = 9
0 + 4y² = 9
y² =


y = ± ( )

Then the y-intercepts are the points (0, ) and (0, )

Just remember: Whichever intercept you're looking for, the other variable gets set to zero.

2.5 Fundamental Theorem of Algebra

This theorem is easy to understand: the highest degree in a polynomial function equates to the total number of zeroes that that function has. Before reading this, most likely you were told to find all the real or rational zeroes of a function. Sometimes the only zeroes a function has are real, sometimes not. Sometimes the zeroes are imaginary.

Here is an example:

The given zeroes of x for f(x) are:

A casual glance at this might make one think that this is, at the very least, a cubic function (a polynomial whose highest degree is 3). But that would be a wrong assumption to make. The presence of the 3-i automatically implies the presence of its conjugate, 3+i. So this function is at least a quartic function, and the zeroes actually are:

The easiest function that could be formed from these zeroes would be:

The easiest thing to do would be to then to multiply the lone x and the (x-2) together, and then to multiply the two imaginary zeroes together:

To deal with the imaginary zeroes, we have two options. We could simply distribute all of the variables in the first imaginary zero to the other, but that is tedious and has a high potential for error. Instead, we will ignore the presence of the x's in the imaginary zeroes:

Then, we merely add up these numbers to get a middle term, and then multiply/foil them to get the last term. To demonstrate:

So now our equation looks like this:

All done! One more thing: a problem asking for all of the x-intercepts and a problem asking for all of the zeroes are not asking for quite the same thing, although they're very, very close. Imaginary zeroes do not appear on a normal x-y graph, so when a problem asks for all of the x-intercepts, it is another way of asking for all of the real zeroes. In other words: you don't need to bother finding the imaginary zeroes. But if the problem wants all of the zeroes, then you have to find all of them, both real and imaginary.

Section 2.1: Polynomial Functions

We will begin discussing this section with the equation that Mr. Wilhelm showed us in class today, which read that a polynomial function is of the form:

There are two unfamiliar variables in play here: a and n.

The coefficients, which are real numbers, are represented in the equation as: 

The value of n must be a nonnegative number. 

n can be:

• a whole number,
• equal to zero
• or a positive integer.

Another important part a polynomial function is its degree. Each polynomial function has a specific degree that is determined by the highest value for n where   

For example, if   , the degree of this function is 2. This is because the x is raised to the second power, which is the highest value for n in this function. 

Let's try another. If , what is its degree? The correct answer is 4. Did I trick you? If you properly foil out this function, you will see that when is distributed to ,  is created! Eureka! Therefore, because has the highest value for n at 4, the degree is 4. 

Now that we know what a polynomial function consists of, lets take a look at what its called. The chart below has three categories, Degree, Name, and Example.

Degree                      Example                       Name

      0                                                                     Constant

      1                                                             Linear

      2                                              Quadratic

      3                                                                   Cubic

      4                                                         Quartic

Now, let's go more in depth into a very widely-used polynomial function, the quadratic. 

The basic formula for a quadratic is , where a, b and c are the coefficients. An example graph of this is:

in which . It is important to note that the quadratic formula is very helpful here:

The two x-intercepts can be found by using the quadratic formula. The intercept farthest to the right will use -b+ while the intercept farthest to the left will use -b-.

This quadratic graph has an axis of symmetry traveling down what would be the line of x=3.

At the bottom tip of the quadratic, located at the point 

(3,-2), lies the vertex. In this case, the vertex of the graph is the minimum. In order to find the x coordinate of the vertex, you can use  . 

Another way to look at the equation of a quadratic polynomial is to use the vertex equation. The vertex equation is:  in which the vertex is simply at (h, k). In the case of the graph above, the vertex equation is 

The last topic that we will discuss is how to change a normal quadratic equation to a vertex equation. A reason we should know this is that it is easier to find the vertex on a graph using the vertex equation than with the normal quadratic equation. The best way to approach this is by using completing the square. 

We will go through this process using four easy steps. 

1. When starting with a quadratic equation, such as , it is important to isolate c. Here, we will isolate 5. So just for now, move the 5 to a far off space on your page. Just forget about it. 

2. Add  to . We are doing this so that we will be able to factor easily later on. Here, we will add 9. But it's not that simple; we can't just add blindly! In order to successfully add 9, we will need to subtract 9 from something else.

3. Remember that 5? You know, the one I told you to forget about? We now need to subtract 9 from the 5 to make it -4. Now, the equation should look like this: 

 


4. The last step is to factor your new . In this case, we will factor so that the new equation looks like .

And thats it! We have successfully changed a quadratic equation into a vertex equation! Hold on, I have one last question for you to make sure you were reading: what is the vertex of the function we just made? The correct answer is (-3, 4).








Thanks for reading!