Chapters

## What are Polynomial Functions?

**common**polynomials.

Linear function | Two monomials | |

Quadratic function | Three monomials | |

Cubic function | 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.

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.

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.

## 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.

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.

Numerator | 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.

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

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

Here are some other simplifications** rules** that can help you:

## 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**.

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 | |

Degree 3 | |

Degree 4 |

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 | Parabola shape is | |

Positive sign in front of a | Parabola shape is |

### 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.

A | B |

## 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.

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**:

Either way, we get the following:

## 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**:

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.

## Step-by-step Example

Let’s graph the following 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:

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

Now, we can **simplify** the terms.

Next, we simplify the** second** rational function.

Putting it all** together**, we get the following.

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