2.7 Graphs of Rational Functions

How to graph a rational function:

1. Find and plot the y-intercept by evaluating f(0).
2. Set the numerator equal to zero and solve to find the x-intercepts.
3. Set the denominator equal to zero to find the vertical asymptotes.
4. Find and sketch the horizontal asymptote if there is one.
5. Use curves to complete the graph between and after the vertical asymptotes.

Example:

Find the y-intercept by plugging in 0 for x.  You then find f(0)=5, therefore making the y-intercept (0,5).
Now to find the x-intercepts set the numerator equal to 0.

0=5+2x
x= -5/2

This tells you that the x-intercept in (-5/2,0)
To then find the vertical asymptote you must set the denominator equal to 0.  From here we can see that the vertical asymptote is line x= -1.
From the information above we can also see that the horizontal asymptote is line y=2

The facts from above allowed us to create this graph of a rational function.

Most of the time with vertical asymptotes, the two parts are on different ends of the asymptote, for example the graph above. Although when you have a vertical asymptote with a multiplicity of two, they are on the same end.

Example:





This has a multiplicity of 2 at the asymptote x=1.
See how in this graph the two parts are at the same end of the asymptote, that is due to the multiplicity of two.

Some graphs can also have holes. When there are the same factors located in the numerator and the denominator of the function it cause a hole at that x value in the graph.

Example:




With this on the graph we should see a hole when x=1 due to the same factor in the numerator and denominator.

Slant Asymptotes:
If the degree of the numerator is one more than the degree of the denominator then the function has a slant asymptote. To find the line of the asymptote you must use long division and divide the numerator by the denominator.

Example:


This divides out to x-2 with a remainder of 2.  The slant asymptote is the line f(x)=x-2.
You can see this in the graph.

2.5 The Fundamental Theory of Algebra

In the complex number system, every nth-degree polynomial function has exactly n zeros. This is derived from the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra - If f(x) is a polynomial of degree n, where n > 0, f has at least one zero in the complex number system.

The fundamental theorem of algebra also gives you the linear factorization theorem.

Linear Factorization Theorem - If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors


Where c1, c2, . . . , cn are complex numbers

Real and Complex Zeros of Polynomial Functions

Example:

 

This polynomial can be factored into:

 

This shows that the zeros are
(x-2) with a multiplicity of two, (x-2i) and (x+2i)

Notice that (x-2) is listed twice and (x-2i) and (x+2i) are both complex zeros.

Remember:
***The n zeros of a polynomial function can be real or complex and they may be repeated***
***Imaginary roots always come in conjugate pairs***

Finding the Zeros of a Polynomial Function

Example: 



The possible rational zeros of this polynomial function are ±1, ±2, ±4, and ±8 
(Find these by taking factors of the constant divided by the factors of the leading coefficient. The leading coefficient is 1 so we only need the factors of the constant)

By graphing this function, you can see that -2 and 1 are both real zeros. 
***Only the real zeros are shown on the graph as x-intercepts***
After finding the real zeros, you can find the imaginary zeros by using synthetic division.
Using -2 and 1 twice you are left with:



To find the zeros we need to factor this





This gives us the remaining non-real zeros. All five zeros of this polynomial are:

1, 1, -2, 2i, and -2i

Conjugate Pairs

In the last example two of the zeros were -2i and 2i. This is because all imaginary roots must come in conjugate pairs, in the form a + bi and a - bi

In a polynomial function with real coefficients if a + bi is a zero of the function and b is not 0 then the conjugate a - bi must also be a zero of the function.

Finding a Polynomial with Given Zeros

Given the degree of a polynomial function with real coefficients and its zeros we can easily determine the polynomial.

Example:

A third degree polynomial function with real coefficients has 2 and 4i as zeros.

Given 4i as a zero we know that -4i must also be a zero. Because the polynomial function is a third degree function we now have all the zeros: 2, 4i, and -4i.

We can write these out as:



To get the original polynomial we can multiply these factors.



This results in:



Factoring A Polynomial

Factors of a Polynomial - Every Polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.

Example:



This can be factored into two quadratic polynomials



This can be further factored to include imaginary numbers



Zeros of a Polynomial Function

Given one complex zero of a polynomial function we can find the remaining zeros.

Example:

5i is a zero of   

Because 5i is a complex number, we know that -5i must also be a zero of this polynomial because all complex zeros come in conjugate pairs.

We can multiply these together to get part of the polynomial:






We can now use long division using this to find the remaining zero.

                                       
                                                                       
/
              
                                                          
                     
                     
                                                                   
                                               

This gives us 



The zeros of this polynomial are:

Section 2.6: Rational Functions and Asymptotes

Section 2.6: Rational Functions and Asymptotes

A rational number is a number that can be written in a ratio.

A rational function is a function that can be written as:

 

In this, N and D are both polynomials.

The domain of f is all numbers that do not make the denominator equal to 0.

 



Example 1.

 Find the domain of

 Solution:   Factor the denominator.   (x+3)(x-3)

            Solve to find values of x that make the denominator = 0,       x=3,-3    

            Domain:  all real numbers except -3 and 3,

Horizontal and Vertical Asymptotes

Definitions of Asymptotes

1. A graph has a vertical asymptote if D(x)=0. This means that all values that are not in the domain are vertical asymptotes.

2. A horizontal line is a horizontal asymptote of the graph if the graph approaches the line as x approaches positive or negative infinity.

The graph will never cross a vertical asymptote, but it may cross a horizontal asymptote

Rules for Asymptotes of Rational Functions

Let f be a rational function:



1. The graph of f has a vertical asymptote at x = a if D(a) = 0.  A vertical asymptote occurs at the value(s) of x that make the denominator equal to 0, which makes the function undefined.

 

2. The graph of f has one horizontal asymptote or no horizontal asymptote, depending on the degree of N and D.

a.        If the degree of the numerator (n) is less than the degree of the denominator (m), then the horizontal asymptote is y=0.

b.        If the degree of the numerator (n) is equal to the degree of the denominator (m), then the horizontal asymptote is quotient of the leading coefficients.

c.          If the degree of the numerator (n) is greater than the degree of the denominator (m), then there is no horizontal asymptote.

3. The graph of f has a slant asymptote if the degree of the numerator (n)   is 1 greater the degree of the denominator (m).

Example 2.

Find any horizontal and vertical asymptotes of the following.

 
 

Solution:   The degree of the numerator is 3, the degree of the denominator is 1.  Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. The vertical asymptote is -8 because -8 makes the denominator equal to 0.

 





 

Solution:   The degree of the numerator is 1 and the degree of the denominator is 1.  Since the degree of the numerator is equal to than the degree of the denominator, the horizontal asymptote is the quotient of the leading coefficient of the numerator and the denominator,  so it is:



Set the denominator equal to zero and solve for x.

            4x-6=0, so the vertical asymptote is:

 

 

 

 

 

Solution:   The degree of the numerator is 1 and the degree of the denominator is 2.  Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0

            Set the denominator equal to zero and solve for x.

            X²-1=0

            X=1,-1,   so the vertical asymptotes are x=1 and x=-1