Rational Functions and Asymptotes

Chapter 2.6 - Rational Functions and Asymptotes

This chapter contains

- Rational Functions
- Vertical Asymptotes
- Horizontal Asymptotes

Rational Functions

Much like a rational number, a rational function can be defined as any function in which both the numerator and denominator are polynomials.  This can be written as the following equation:



The domain of the function is all real number, excluding any inputs that would result in D(x) equaling 0; this would create a vertical asymptote.

Vertical Asymptote

A vertical asymptote is formed when the denominator function, or D(x), is equal to zero.  When this occurs, the output, or f(x), becomes undefined.  When you graph this, you will notice that the graph will run along the vertical asymptote, up to infinity and negative infinity.

In order to find a vertical asymptote, you have to set D(x) equal to zero, solve out, and at the zeroes set vertical asymptotes in the graph.

Example:

Horizontal Asymptotes

A horizontal asymptote is a y coordinate in which the function cannot cross.  The graph will get infinitesimally get closer to the asymptote in both the negative and positive infinity, but it will never reach it.  A horizontal asymptote can be crossed at certain points (see figure below), but in the infinite and negative infinite directions it will not.

How to find the horizontal asymptote:

If the degree of the numerator is less than the denominator:
In this case, the horizontal asymptote will always be zero, but it is still possible for the graph to cross through zero.  In the above example, the horizontal asymptote is 0, because the degree of the numerator is lower than the degree of the denominator.

If the numerator and denominator have equal degrees:

In this case, the horizontal asymptote is determined as follows:
if (only including the highest degrees)  
then the asymptote will be at 

If the degree of the numerator is greater than the degree of the denominator:

There is no horizontal asymptote.

The FUNdamental Theorem of Algebra

In this section we will discuss the Fundamental Theorem of Algebra, the Linear Factorization Theorem,  Complex Zeros, and much more.

Definitions

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

Linear Factorization Theorem: If f(x) is a polynomial of a degree n where n > 0, f has precisely n linear factors. f(x) = f(x)=a_n(x-c₁)(x-c₂)…(x-c_n) where c_1, c_{2, .....} c_{n ,  are complex numbers.}

_{Application and Examples}

_{To find the zeros of an equation one must first understand what a zero is. This is a point on the graph of any function where the line of the function crosses the x-axis. A zero can be found by plugging 0 in for the y value of a function and then solving for x.} 

Example:

f(x) = x-2

+2      +2

x = 2

For this equation the graph will cross the x-axis at the point (2,0)

Most functions will not be as easy to solve as the previous example. For these it is recommended that for smaller functions such as cubic or quadratic that one attempts factoring or the quadratic equation first.

Factoring Example:

x²-6x+9=f(x)

(x-3)(x-3)=f(x)

This equation has two zeros x=3 and x=3.

For larger functions containing exponents much larger than these shown a useful method to use would be synthetic division, which was demonstrated in an early section.

The zeros will not always be easy real numbers. Many equations will contain imaginary numbers for zeros. Before we discuss complex numbers we have to know what they are.

Imaginary Numbers- This is a sub category of complex numbers which includes numbers that cannot be expressed with out the use of non-real terms such as i. Examples: 3+i, i, 3-i

Equations that involve imaginary numbers will appear as though the whole graph has been shifted up or down along the x-axis. For these types of graphs imagine sliding the x-axis upwards and see if there is a point at which the is the same amount of x-intercepts as there should be based upon the highest exponent value.

When there are complex numbers as zero which involve in some way an imaginary number their conjugate will also be a zero. If a+bi, where a and b are real numbers, is a zero, than a-bi will also be a zero.

Never will there be one of the values for a conjugate pair be a zero and the other not be a zero if the equation is truly a function.

When Imaginary numbers are found in a graph they will appear in some form of this:

Real zeros on the other hand will appear as intercepts such as these: 

A graph with both real and non-real zeros may appear like this:

Working From Zeros Back to the Original Equation can be a long tedious process though never should it be too difficult.

Example: the zeros for a function are 2,3, and 7. Write the equation for this function as a polynomial.

(x-3)(x-2)(x-7)=f(x)

(x²-5x+6)(x-7)=f(x)

x³-12x²+41x-42=f(x)

To solve problems like this simply take the zeros and place them as you would find them at the end of factoring a polynomial. Then work backward as shown and FOIL the linear functions until they are all one large polynomial function again. 

2.3 Real Zeroes of Polynomial Functions

This section contains information on long division of polynomials, synthetic division, the remainder theorem and the rational zero test.

Long Division of Polynomials

One way to divide polynomials by other polynomials is to use long division.

Example: Divide by x+3

After dividing, we can tell that the remainder is zero because there are no numbers that are left over. Since we know (x+3)(2x+4)=we can find the x intercepts of the graph:

(x+3) (2x+4)

x+3=0   2x+4=0

x=-3      x=-4/2

              x=-2

The x intercepts are located on the graph at x=-3 and x=-2.

When dividing polynomials, sometimes there will be a remainder that is left over.

Example: Divide by x+2

In this equation, the remainder is -23. The answer to this problem can be written as:

Synthetic Division

Synthetic division is another method you can use when finding the zeroes of polynomials.

Example: Divide  by x+3

Setting it up: 1. The divisor is x+3 which means that x=-3. The -3 is placed to the left.

                    2. The coefficients of the dividend are 2, 10 and 12. These numbers are placed in order to the                            right of the -3.

Step 1: Bring down the 2

Step 2: Multiply the 2 by -3= -6

Step 3: Add 10 + (-6) = 4

Step 4: Multiply the 4 by -3= -12

Step 5: Add 12 + (-12) = 0

When using synthetic division, multiply the terms going diagonally and then add the terms vertically. x+3 is a factor of the equation because it has no remainder. If there was a remainder, then x+3 would not be a factor.

Example: Divide  by x+5

Since the remainder is 12, x+5 is not a factor.

The Remainder Theorem

The remainder theorem: f(k)=r

If a polynomial f(x) is divided by x-k

Example: Divide  by x+6

Using the remainder theorem we know F(-6) =-65

This tells us that (-6, -65) is a point on the graph of f which can be seen below.

Rational Zero Test

If this polynomial

Has integer coefficients, every rational zero of f can be found using this equation:

p= factors of the constant term 

q= factors of the leading coefficient

Example:  Find all possible rational zeroes of 
p= factors of 6
q= factors of 5

*These are only the possible rational zeroes. They can be tested individually using synthetic division to determine if they are actual rational zeroes.
Example: Test x=-1 to see if it is a rational zero

Since there is a remainder of -4, x=-1 is not a rational zero.