## Graphing Polynomial Functions

#### Aligned To Common Core Standard:

**High School Interpreting Functions** - HSF-IF.C.7c

How to Graph Polynomial Functions -
Suppose that you have been given data of millions of dollars of a fictional company for six years in a table, for example:
Year: 2006 | Revenue = 52.4 | Year: 2007 | Revenue = 52.8 | Year: 2008 | Revenue = 51.2 | Year: 2009 | Revenue = 49.5 | Year: 2010 | Revenue = 48.6 | Year: 2011 | Revenue = 48.6 | Year: 2012 | Revenue = 48.7 | Year: 2013 | Revenue = 47.1
This revenue can easily be indicated using a polynomial function: R(t) = -0.037t^{4} + 1.414t^{3} - 19.777t^{2} + 118.696t - 205.332
In this case, R represents the revenue generated by the company in millions and t is a representation of the years which equals 6. So over which years the revenues started decreasing or dropping?
Turning points and multiplicity
Graphs don't behave the same as an x-intercept. Sometimes a graph with cross the x-axis at an intercept. Other times it might even bounce off.
Suppose that we have a graph of the function:
f (x) = (x + 3) (x - 2)^{2} (x + 1)^{3} A series of worksheets and lessons that help students learn to bring polynomial functions to life on a graph.

### Printable Worksheets And Lessons

- Three Part Question Step-by-step Lesson- Sketch the graph, find the maximum number of turns, and list all the possible zeros.
- Guided Lesson - We run the same play on some more polynomials. What is the maximum number of turns a graph of this function could make?
- Guided Lesson Explanation - I tried to give you a little outline for problems like this in the beginning.
- Practice Worksheet - Each question asks you a slightly different thing, so make sure to read it.
- Matching Worksheet - Which letter matches the problem at each turn? The way in which the questions are asked can indicate to you the possible line of answering.

#### Homework Sheets

Finding the maximum and minimums are truly the most useful skill here.

- Homework 1 - n is each to the degree of the polynomial. Polynomial functions have the maximum number of turns describe as n - 1.
- Homework 2 - Sketch a graph of this function. What is the maximum number of turns a graph of this function could make?
- Homework 3 - List all the possible real zeros. Plot the points on the graph.

#### Practice Worksheets

Listing all the possible real zeroes has a great application in waste treatment. I took a field trip on it.

- Practice 1 - Plop this on a graph please. This will help you make sense of them.
- Practice 2 - Find the final value of the following problem.
- Practice 3 - Use the Rational Zero Theorem to list all the real zeros.

#### Math Skill Quizzes

The quizzes most focuses on sketching the graphs.

- Quiz 1 - What is the maximum number of turns a graph of this function could make? State the number of real zeros.
- Quiz 2 - Sketch the graph of x
^{2}+ 3x + 8. - Quiz 3 - Interpret and sketch these graphs. You will also be asked to state the number of real zeros that are present.

### Common Features of Graphs of Polynomial Functions

Polynomial functions are your standard function that have an aspect that includes a non-negative integer power of x. When we plot polynomial functions on a graph, they are constant and linear, so they always form straight lines. As the exponential portion of the functions increase in value, the lines begin to bend and form curves. These exponential portions and referred to as the degree of the function. As the degrees of the functions gets larger, those curves begin to have steeper curves. When you start to look at these graphs you will notice the concept of turning points. This is sections of the graph where the slope may drastically change upward or downward that results in a bend in the graph. When we graph these functions, we will note that they are continuous and have no breaks within them. We should also note that any real number is a valid input for these functions. We can often find the x-intercept of these functions by setting them equal to zero and solving for the possible values of x. Which makes sense because the x-intercept is exactly where x equals zero. You can also find the x-intercept by factoring. Remember if you calculate that x has several values that can be where the line curves and touches the intercept multiple times.