Lesson 1

Lesson 1: Polynomial Functions and Their Graphs

 

Focus

 

Comstock/Thinkstock

 

Mathematical models can be useful for making predictions. Consider the following scenario:

 

The treasurer of a drama club is putting together a budget for the club’s next production. She has determined the revenue the club can expect to generate for its next production as a function of ticket price. She has also determined the cost of the production, also as a function of ticket price. The following graph shows her predictions.

 

How could the treasurer determine the ticket price that would produce revenue greater than costs? One way would involve determining the equation of a function passing through the production cost data points and the equation of another function passing through the revenue data points.

 

In this lesson you will learn about different classes of polynomial functions—their characteristic shapes and algebraic forms.

 

Lesson Outcomes

 

At the end of this lesson, you will be able to

• sketch graphs that represent constant, linear, quadratic, and cubic functions
• relate characteristics of graphs of polynomial functions to the functions’ algebraic forms

Lesson Questions

 

You will investigate the following questions:

• How are the characteristics of the graph of a polynomial function related to the degree of the polynomial?
• How are the characteristics of the graph of a polynomial function related to the leading coefficient and constant term?

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 1 Assignment (Download the Lesson 1 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you are having difficulty with concepts or calculations, contact your teacher.

 

Remember that these questions and activities provide you with the practice and feedback that you need to successfully complete this course. You should complete all the questions and place the answers in your course folder. Your teacher may wish to view the work to check on your progress and to see if you need help.

 

Time

 

Each lesson in Mathematics 30-2 EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

 

This time estimation does not include time required to complete Going Beyond activities or the Module Project.

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

 

Before beginning this lesson, you should be able to

• use function notation
• create tables of values, plot points, and graphs
• determine domain and range
• analyze linear functions and their graphs
• analyze quadratic functions and their graphs
• determine x- and y-intercepts

Are You Ready?

 

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1.  
   1. Given the function f(x) = 3x − 6, find
      1. f(−1)
      2. f(0)
      3. f(3)
      4. x, when f(x) = 0
      
      Answers
   2. State the answers from question 1.a. as sets of ordered pairs. Answers
   3. Graph f(x) without technology. Answers
   4. State the domain and range for f(x). Answers
   5. State the x- and y-intercepts for f(x). Answers
   6. The slope y-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. State the slope of f(x). Look at your answer to question e. for the y-intercept: Is it the same as the last number in the equation? Answer
2.  
   1. Given the function g(x) = 2x2 − 4x − 6, find
      1. g(−2)
      2. g(0)
      3. g(4)
      4. x, when g(x) = 0
         
         
      Answers
   2. State the answers from question 2.a. as sets of ordered pairs. Answers
   3. Graph g(x) without technology. Answers
   4. State the domain and range for g(x). Answers
   5. State the x- and y-intercepts for g(x). Answers
   6. The function g(x) is a quadratic function and can be written as g(x) = 2(x − 1)2 − 8.
      1. What is the vertex?
      2. What is the axis of symmetry?
      3. Describe whether the graph opens up or down.
      4. Is the graph wider or thinner than g(x) = x2? What number in the equation can be used to predict this?
      5. What is the minimum value?
      6. What are the domain and range?
      
      Answers

If you answered the Are You Ready? questions without difficulty, move to Discover.

 

If you found the Are You Ready? questions difficult, complete the Refresher.

Refresher

 

To be successful in creating graphs of functions, you need to know how to plot coordinates. If you feel you need a review of this, go to the Coordinates applet.

Instructions for use: Grab the point in the demonstration applet, and observe how the coordinates change as you move the point around. You may want to choose “show indicator lines” to remind yourself how the coordinates are determined.

---

For a review of domain and range, use the Domain and Range Compressor applet.

 

 

 

 

 

 

 

Instructions for use:

• Choose one of the graph types.
• To see the corresponding domain, drag the Domain Shadow slider all the way to the bottom. The corresponding domain shadow (i.e., the blue line along the x-axis) represents the domain of the function.
• To determine the corresponding range, drag the Range Shadow slider all the way to the left. The corresponding range shadow (i.e., the red line along the y-axis) represents the range of the function.

---

Use the “Quadratics in Vertex Form—Activity A” gizmo to explore and investigate quadratic functions in f(x) = a(x − h)2 + k form. Click on the “Exploration Guide” and follow the directions to get the full use of this interactive multimedia piece.

---

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

The drama club scenario presented in the Focus requires that you recognize shapes of scatter plots and relate them to polynomial functions. The following activity will help you recognize the shapes for constant, linear, quadratic, and cubic polynomial functions.

 

Try This 1

 

Open Polynomial Data Fitter. Your goal is to drag the blue points to make the line or the curve match the red points. If you can’t make the line or the curve match the red points, click one of the buttons (“Constant,” “Linear,” “Quadratic,” or “Cubic”) and try moving the blue points again.

 

Once you have matched the first set of red points, move the “Data Set” slider to select another set of red points to match.

 

---

Try This 1 gave you experience with the shapes of the graphs of constant, linear, quadratic, and cubic functions. You also saw the different equations for the graphs you were manipulating.

 

Share 1

 

With a partner or in a group, come to agreement on the typical shape, highest exponent, and sample equation for each type of function. You may want to organize your results in a table similar to the following. The entries for linear functions have been filled in for you.

 

Type of Function	Highest Exponent	Example Equation	Example Shape
Constant
Linear	1	f(x) = −5x + 3
Quadratic
Cubic

 

If required, save a record of your discussion in your course folder.

Explore

 

The shapes of the functions you experimented with in the Discover section are summarized in the following table.

 

Type of Function	Highest Exponent	Example Equation	Example Shape
Constant	0	f(x) = −2
Linear	1	f(x) = −5x + 3
Quadratic	2	f(x) = x2 + 2x − 3
Cubic	3	f(x) = 2x3 + 5x2 + 2x + 1

 

The functions you graphed in the Discover are all examples of polynomial functions.

 

 

In Discover you specified the highest exponent (on a variable). This is called the degree of the polynomial. In this module you will work with degree 0, 1, 2, and 3 polynomial functions.

 

Degree	Name	Example
0	constant	f(x) = 3.5
1	linear	f(x) = 4x − 1
2	quadratic	f(x) = 3x2 + 6x − 1
3	cubic	f(x) = 5x3 − 4x2 + 7x + 3

 

Try This 1 introduced you to the general shapes and characteristics of polynomial functions. When describing a graph, it is often useful to consider the following characteristics:

• x-intercepts
• y-intercepts
• end behaviour
• domain
• range
• turning points

Most of these characteristics should be familiar to you; however, end behavior and turning points may be new.

 

The end behaviour of a graph describes what happens as the x-values become very large positive or very large negative numbers. In this course, you will typically describe end behaviour by stating the quadrant the graph is in for large negative x-values and large postive x-values.

 

 

A turning point occurs when a graph changes from increasing to decreasing or from decreasing to increasing.

 

 

Try This 2

 

Open Polynomial Function Explorer.

 

 

 

Your goal is to determine the following:

• number of possible x-intercepts
• number of possible y-intercepts
• end behaviour
• domain
• range
• number of possible turning points

Drag the blue points to explore the shapes of each type of function. (Click the buttons to change the function type.) It may be helpful to choose “Show x-intercepts” and “Show y-intercept.”

 

A table like the one that follows will help you organize your results.

 

Type of Function	Constant	Linear	Quadratic	Cubic
Degree	0	1	2	3
Example Function	f(x) = a	f(x) = ax + b	f(x) = ax2 + bx + c	f(x) = ax3 + bx2 + cx + d
Possible Number of x-intercepts
Possible Number of y-intercepts
End Behaviour
Domain
Range
Number of Possible Turning Points

Adapted from PRINCIPLES OF MATHEMATICS 12 by Canavan-McGrath et al. Copyright Nelson Education Ltd. Reprinted with permission.

 

Save your responses in your course folder.

In Try This 1 you saw that the general shape of a graph can be predicted by the degree of the function. In Try This 2 you used the degree of a graph to predict some of the details of the graph.

In the first part of this lesson, you related the degree of polynomial functions to the shapes of the corresponding graphs. In the next part of this lesson, you will investigate the specific components of the equations that affect the corresponding graphs.

 

Try This 3

 

Use the End Behaviour Explorer to look for patterns in end behaviour and the y-intercepts. When the applet starts, a first degree (linear) function is shown. Change the “a” and “b” sliders, and pay attention to how the

• end behaviour of the function changes
• y-intercept changes (Be sure to check the “Show y-intercept” box.)

Repeat for degree 0, 2, and 3 polynomial functions.

 

1. Which slider(s) affect the end behaviour of the graphs? Does the sign of this slider affect the end behaviour?
2.  
   1. Which slider(s) affect the y-intercept of the graphs?
   2. Record your observations in a table like the following.
      
      

      Degree	Type of Function	End Behaviour
      		Starting Quadrant	Ending Quadrant
      0	constant
      1	linear
      2	quadratic
      3	cubic

Save your responses in your course folder.

 

Share 2

 

With a partner or in a group, share your end behaviour observations from Try This 3. Once you come to agreement on end behaviour, discuss the following questions.

1. Which parts of the polynomial can be used to predict end behaviour?
2. Which parts of the polynomial can be used to predict the y-intercept?

If required, place a summary of your discussion in your course folder.

In End Behaviour Explorer that you used to complete Try This 3, you may have found that only the leading coefficient, the a-value, affected the end behaviour. You should have also discovered that the end behaviour only changed if the sign of the leading coefficient changed.

 

The sign of the leading coefficient predicts end behaviour.

 

Cubic	y = ax3 + bx2 + cx + d
Quadratic	y = ax2 + bx + c
Linear	y = ax + b
Constant	y = a

 

 

 

In Try This 3 you may have also found that the y-intercept has the same value as the constant term.

 

The value of the constant term predicts the y-intercept.

 

Cubic	y = ax3 + bx2 + cx + d
Quadratic	y = ax2 + bx + c
Linear	y = ax + b
Constant	y = a

 

Connect

 

Lesson 1 Assignment

Lesson 1 Summary

 

In this lesson you learned the relationships between polynomial degree and the following characteristics of the corresponding graph:

• x-intercept
• end behaviour
• domain
• range
• turning points

You also learned how the polynomial’s leading coefficient and constant term affect the end behaviour and the y-intercept, respectively.

 

In Lesson 2 you will learn how to determine lines of best fit for data points.