What are Polynomial Functions?

Rational functions are made up of polynomials. In order to start graphing rational functions, you should first get a good understanding of what polynomial functions are. Let’s start by taking a look at some very common polynomials. 
 

Linear function y = mx + b Two monomials
Quadratic function y = ax^{2} + bx + c Three monomials
Cubic function y = ax^{3} + bx^{2} + cx + d Four monomials

 

You may have noticed the term ‘monomial.’ A monomial is the basic building block of polynomials. Let’s take a look at the different types of monomials.

 

x 3 2x 2xy
One variable One constant One variable times one constant One constant times two or more variables

 

When we combine monomials, we can form the polynomial equations form before.

polynomial_binomial

As long as we have one or more monomials, we have a polynomial equation. However, polynomial equations with one, two and three monomials also have special names. We know monomial, and in the image above we can see: binomial and trinomial.

 

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Rational Functions

Rational functions are defined as two polynomial functions divided by each other. In other words, we have one polynomial function in the numerator and one polynomial function in the denominator.

rational_function

Remember that a polynomial is any function with one or more monomials. This means that technically, all polynomial functions are also rational functions. This is because any polynomial function can be re-written to be divided by 1. Take a look at the image below.

polynomial_rational_function
Numerator 3x + 4 Binomial
Denominator 1 Monomial

 

When we are given a rational function, the first thing we need to do is simplify it. Simplification means that we are grouping all variables together and all constants together as much as possible. Let’s take a simple example.

rational_simplification

If you remember, when we have a fraction, we can transform it into the following.

simplify_denominator

This can often be the first step in simplifying rational functions. Let’s apply it to our example.

rational_function_example

Now that we have simplified the rational function, it is a lot easier to graph.

 

x y
1 7
2 5
...
10 3.4

 

linear_function_rational

 

Here are some other simplifications rules that can help you:

simplification_quadtratic_cubic

 

Common Polynomials

Now that we have gone through all the building blocks of rational functions, we can begin to discuss how to graph rational functions. Since the goal of simplifying rational functions is to reduce them to a polynomial, we will start with graphing the most common polynomials.

 

Linear Polynomial

As mentioned, the linear polynomial is the easiest polynomial to graph. We only have four elements in a linear polynomial.

 

y Output Dependent variable
m Slope The rate of change
x Input Independent variable
b y-intercept Value of function if x = 0

 

To graph a linear polynomial, simply plug in values of x in order to get the resulting y value. No matter what equation or x values you choose, a linear polynomial will always result in a straight line.

rational_functions
A B C
y = 4x + 3 y = -5x + 2 y = 40x+1

 

Quadratic

A quadratic equation has a degree of two. Degrees in polynomial functions mean the highest value of power in the equation.

 

Degree 2 y = ax^{2} + bx + c
Degree 3 y = ax^{3} + bx^{2} + cx + d
Degree 4 y = ax^{4} + bx^{3} + cx^{2} + dx + e

 

In order to graph a quadratic equation, you should simply plug in values of x and plot the resulting y. The shape of a quadratic equation will always be a parabola. One easy trick to remember is:

 

Negative sign in front of a y = -ax^{3} + bx^{2} + cx + d Parabola shape is \cap
Positive sign in front of a y = ax^{3} + bx^{2} + cx + d Parabola shape is \cup
quadtratic_functions_negative_positive

Cubic

A cubic function has a degree of 3. Just like the other polynomials we’ve talked about, graphing a cubic function involves picking values for x and graphing the resulting coordinates. A cubic function will always result in an ‘S’ shape.

 

Just like with a quadratic function, depending on the sign in front of the ‘a’ term, this ‘S’ shape can either be normal or backwards. Take a look at the examples below.

cubic_function_positive_negative
A B
y = ax^{3} + bx^{2} + cx + d y = -ax^{3} + bx^{2} + cx + d

 

Graphing Simple Rational Functions

Now that we have a good grasp on all the components that go into graphing rational functions, let’s start by graphing simple rational functions.

rational_function_explained

There are two different polynomials in the denominator and numerator:

 

A Binomial Linear function
B Monomial Linear function

 

There are two ways we can do this, simply plug in values for x and plot the coordinates:

 

x y
1.0 7.0
2.0 4.5
3.0 3.7

 

Or simplify the equation first:

rational_simplification_example

Either way, we get the following:

rational_function_graphed

 

Graphing Complex Rational Functions

When you’re graphing more complex rational functions, you can follow either of the two processes we mentioned above. Let’s take an example:

rational_function_two_functions

The polynomials here are:

 

A Trinomial Quadratic function
B Binomial Linear
C Binomial Linear

 

We can plug in values of x:

 

x y
1 8
2 12
... ...

 

However, this can take too long with more complex rational functions. So instead, we prefer to simplify first.

simplification_quadtratic

 

graph_rational_function

 

Step-by-step Example

Let’s graph the following rational function:

cubic_rational_function

First, in order to simplify this rational function, we first look for any patterns. Let’s take the two elements of the equation separately. Starting with the first one, we have one trinomial and one binomial. Let’s factor the first one:

cubic_rational_function_example

The binomial follows a perfect square pattern. This means we can factor it in the same way.

cubic_rational_function_simplification

Now, we can simplify the terms.

simplification_radical

Next, we simplify the second rational function.

simplification_same_denominator

Putting it all together, we get the following.

rational_function_simplified

 

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.