Course Profile Advanced Functions and Introductory Calculus (MCB4U), Grade 12, University Preparation, Combined

Unit 1: Graphs of Polynomial Functions

Time: 13 hours

Unit Description

Students extend their knowledge of linear and quadratic functions to general polynomials. An emphasis is placed on making connections between the equations and the graphs of polynomials with the nature of change being a crucial underlying theme. Students work with functions represented in a variety of forms, such as graphical, numerical, algebraic, or verbal, and develop an understanding of the connections between these forms. A thorough treatment of symmetry, zeros, and behaviour around intercepts and at the ends provides students with a comprehensive set of analysis tools to draw upon throughout the course.

Activity 1.1: The More It Changes…

Time: 2 hours

Description

Students use finite differences to describe the rates of change of polynomial functions and explore how these differences relate to the equation and the graph of the function.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written materials effectively;

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4b - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability.

Strand(s): Advanced Functions

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees.

Specific Expectations

AF1.02 - describe the nature of change in polynomial functions of degree greater than two, using finite differences in tables of values;

AF1.03 - compare the nature of change observed in polynomial functions of higher degree with that observed in linear and quadratic functions.

Prior Knowledge & Skills

· constructing tables of values and calculating finite differences

· determining the degree of a polynomial and identifying functions as linear or quadratic

Planning Notes

· Students need graph paper to record sketches.

· If spreadsheets are used, a computer lab or portable computer and projection device are required.

Teaching/Learning Strategies

Teacher Facilitation

The teacher needs to be precise when making such statements as “a linear function grows at a constant rate,” “a quadratic function grows at a constantly increasing rate,” etc. By the end of the activity, students should have a thorough understanding that varying the degree of the polynomial affects “how” the polynomial changes, which is what determines the shape of the corresponding graph. Rate of change of the rate of change should be introduced, using the example of distance, velocity, and acceleration. Emphasis is placed on the fact that any successive column in the difference table is the rate of change of the previous one (if students can interpret one column, theoretically they can handle them all). For a linear function, the height (y-value) changes at a constant rate; for a quadratic function, the slope changes at a constant rate; for a cubic function, the rate of change of the slope changes at a constant rate.

Part A should not take very long as students have seen these functions before.

In Part B, the teacher may leave the relationship between the leading coefficient (a) and the constant difference (CD) as an extension, depending on the interests and abilities of the class.

Teacher Notes

This activity could be adapted to a teacher-led activity using spreadsheet software or equivalent features on a graphing calculator; an overhead projection device would be helpful in this case. Alternately, a template can be made available on the computers to facilitate the discovery aspect of the activity.

Student Activity

In this activity, we use finite differences to determine the type of function being examined, how the finite differences relate to how a function changes, and how this affects the shape of the corresponding graph. In the early 1900s, finite differences were used by Ada Lovelace and Charles Babbage in one of the earliest calculator/computers, known as the difference engine. The purpose of the difference engine was to generate tables for complex functions using the properties of finite differences. It was the only way the functions, used for navigation, engineering, and astronomical computations, could be evaluated at the time.

Part A

For this part of the activity, the equations of the functions are known. Generate tables and graphs and investigate how the finite differences change as the degree of the polynomial increases.

1. Consider the function y = x.

a) What type of function is it?

b) Complete the table of values. c) Calculate the first differences.

x	y	1st Differences
-3		> > > > > > >
-2
-1
0
1
2
3

d) Sketch the graph on grid paper.

2. What do the first differences tell us about the relationship between x and y? Include a discussion of rates of change in your answer.

3. In this case, the first differences were positive. How would the graph differ if the first differences were negative?

4. Consider the function y = x².

a) What type of function is it?

b) Complete the table of values. c) Calculate the first and second differences.

x	y	1st Differences	2nd Differences
-3		> > > > > > >	> > > > > > >
-2
-1
0
1
2
3

d) Sketch the graph on grid paper.

5. What do the first differences tell us about the relationship between x and y? (Is it a linear relation?) What do the second differences tell us? Again, include a discussion of rates of change in your answer.

6. In this case, the second differences were positive. How would the graph differ if the second differences were negative?

7. Consider the function y = x³

a) What type of function is it?

b) Complete the table of values. c) Calculate the first, second, and third differences.

x	y	1st Differences	2nd Differences	3rd Differences
-3		> > > > > > >	> > > > > > >	> > > > > > >
-2
-1
0
1
2
3

d) Sketch the graph on grid paper.

8. What do the first differences tell us about the relationship between x and y? What do the second differences tell us? (Is the function quadratic?) What do the third differences tell us? Include a discussion of rates of change in your answer.

9. The third differences were positive. How would the graph differ if the third differences were negative?

10. Predict what you would discover if you investigated the finite differences for the function y = x⁴ and describe the relationship between x and y in terms of rates of change. Be as specific as possible.

Part B

A spreadsheet is an excellent tool to create and manipulate a “numerical model” for investigation of the nature of change in polynomial functions. Generate a spreadsheet to calculate finite differences. The formulas required are shown to the right. The function being examined is f(x) = x⁴ – 4x² (note that the 4th differences are constant).

Cell

Formula

= A4 + 1

= A4 ^ 4 – 4 * A4 ^ 2

= B5 – B4

= C6 – C5

= D7 – D6

= E8 – E7

Fill the formulas (in B4, C5, D6, E7, and F8) down as far as you like.

Many modifications can enhance this spreadsheet. For example, 5th and 6th differences could be added and, since the formula in B4 would have to change every time you used a different function, you could create a more advanced spreadsheet to calculate finite differences for general polynomials. This is a great way to explore how changing the various coefficients affects the finite differences and the patterns found in the data. If the equation is unknown, finite differences can be an invaluable tool in determining it.

1. a) State the type and degree of the function that is represented by each equation.

b) Use a spreadsheet to verify that the appropriate differences are constant; record the differences. The first one is done for you.

y = x³ – 2x + 1	Cubic (3rd diff = 6)	y = 3x³ – 2
y = x² – 3x – 7		y = -5x² + 5x + 1
y = x – 7		y = 7x – 5
y = -2x³ + 5x² + 8		y = 4x² + 9x + 3
y = -3x – 8		y = 5x³ – 2x² + x – 4
y = 2x² – 5x		y = -10x + 11

2. Based on your findings, it should be apparent that similar results occur for linear, quadratic, cubic, and other polynomial functions. Assuming this to be true, complete the following statements:

· The function y = 3x³ – 2x² – 1 is a cubic function; the third differences will be constant.

· The function y = 13x – 11 is a __________ function; the ______ differences will be constant.

· The function y = –2x² – 5x – 3 is a ______ function; the ______ differences will be constant.

· The function y = 8x⁴ – x is a ___________ function; the ______ differences will be constant.

· The function y = 2 – 3x² – 7x⁵ is a _____ function; the ______ differences will be constant.

3. What types of functions are represented by the following sets of data?

a)	t	1	2	3	4	5	b)	x	1	2	3	4	5	c)	n	1	2	3	4	5	6
	h	2	9	22	41	66		y	0	8	36	96	200		c	53	6	1	-4	-3	58

Finite differences can also be used to determine the equation of polynomial functions. Consider the points belonging to a function (from Grade 9).

x	y
2	10
3	13
4	16
5	19

The first differences are constant, meaning the function is linear (y = ax + b). Since the constant difference is 3, a = 3. Also, by working backwards, the previous point must have been (1, 7) and the one before that (0, 4), meaning b = 4. The equation of the function is y = 3x + 4.

Obviously, the complexity increases as the degree increases, but a substantial amount of information can be discerned by using finite differences. Notice that the first differences were 3 and the leading coefficient (a-value) is 3. Do you think this is always the case? What about a quadratic function?

4. To help determine the relationship between the leading co-efficient (a-value) and the constant difference in general use a spreadsheet to complete the table and watch for patterns to emerge. [Hint: Many of these equations appeared in #1 – save yourself some work where possible].

Linear Function	CD	Quadratic Function	CD	Cubic Function	CD
y = x – 7		y = x² – 3x – 7		y = x³ – x – 7
y = 2x – 3		y = 2x² – 5x		y = 3x³ – 2
y = -3x – 8		y = 4x² + 9x + 3		y = -2x³ + 5x² + 8
y = 7x – 5		y = 0.1x² + 4		y = 5x³ – 2x² + x – 4
y = -10x + 11		y = -5x² + 5x + 1		y = 0.1x³ – 2x²

5. a) Is there a relationship between the leading coefficient (a) and the constant difference (CD)?

b) How does it change as the degree of the polynomial increases?

c) What would the 4th differences for y = 3x⁴ – 7x + 1 be? (Check using a spreadsheet)

6. For the tables of values in #3, determine the leading co-efficient and the constant term for the corresponding functions.

Teacher Facilitation

The teacher guides students towards the discovery that:

· if 1st differences are constant, the polynomial is degree 1 and a = CD;

· if 2nd differences are constant, the polynomial is degree 2 and or ;

· if 3rd differences are constant, the polynomial is degree 3 and or ;

· if 4th differences are constant, the polynomial is degree 4 and or ;
where a is the leading coefficient and CD is the constant difference that eventually arises.
This is valid only if the x-values are going up by 1. Otherwise ; it may not be productive to go into this much detail.

Also, students should realize that you can back-up the function, using finite differences, to the point where x = 0 (the y-intercept), and the constant value in the equation is determined.

Follow-up Questions

Students should be assigned questions similar to #3, #5c, and #6 from Part B.

Extensions

· After the activity, the teacher could lead a discussion regarding functions that never yield a set of constant finite differences, e.g., y = 2^x. This would hopefully help students realize that polynomial functions are special in that they will eventually yield a column of constant differences.

· Determining the complete equation of a polynomial of degree 3 or higher using finite differences is an enrichment activity that lends itself to higher-order thinking and good mathematical problem solving.

Assessment & Evaluation of Student Achievement

· Application – As formative assessment in the form of a short quiz, students determine the type of function represented by a set of data (by calculating finite differences).

· Thinking/Inquiry/Problem Solving – Students determine the type of function given a table of values containing x-values that go up by different increments or are out of order. Students work backwards to determine other values in the chart that are not given, such as the y-intercept.

Accommodations

· Students who are unfamiliar with spreadsheets could be paired with students possessing computer expertise. Alternately, the spreadsheet template could be made available to students, requiring them only to enter the coefficients as necessary.

· Students with particular aptitude could be challenged to write a computer or graphing calculator program to perform the same role as the spreadsheet.

Activity 1.2: Shaping Up!

Time: 1.25 hours

Description

Students investigate various properties and behaviours of polynomial functions by comparing the graphs of these functions using technology. Students predict the general shape of the graphs of polynomial functions of varying degrees by analysing end behaviour and symmetry.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision making, problem-solving, time, and resource management skills;

CGE7j - a responsible citizen who contributes to the common goal.

Strand(s): Advanced Functions

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees.

Specific Expectations

AF1.01 - determine, through investigation, using graphing calculators or graphing software, various properties of the graphs of polynomial functions (e.g., determine the effect of the degree of a polynomial function on the shape of its graph; the effect of varying the coefficients in the polynomial functions; the type and the number of x-intercepts; the behaviour near the x-intercepts; the end behaviours; the existence of symmetry);

AF1.03 - compare the nature of change observed in polynomial functions of higher degree with that observed in linear and quadratic functions.

Prior Knowledge & Skills

· familiarity with the graphs and properties of linear and quadratic functions

· use of a graphing calculator and/or other graphing software

Planning Notes

· Students need access to a graphing calculator or graphing software, as well as graph paper.

Teaching/Learning Strategies

Teacher Facilitation

The teacher reviews the general form of a polynomial
[(f(x) = ax + b, f(x) = ax² + bx + c, f(x) = ax³ + bx² + cx + d)] and defines related terms, such as polynomial function, leading coefficients, constant term, and degree of a polynomial. The teacher should make students aware that they will be learning different properties of graphs throughout the course; this activity adds some tools to their graph analysis toolbox. The window settings on the calculator are important and may need to be adjusted. The teacher may need to guide the discussion around end behaviour [i.e., what happens to the function for extreme values of x (or )] using the idea of the dominant term. The teacher can introduce the notation “” to provide students a means of communicating these ideas. In Parts C and D, the calculations may get tedious; students may work in small groups and the results can be compiled on the board or an overhead. Alternately, the TABLE features of a graphing calculator can expedite calculations and enable students to focus on the graphical implications of the activity. In Part C, it isn’t likely that many students will come up with the phrase “symmetric about the origin.” The teacher will want to draw this out. In addition, the teacher may facilitate a discussion regarding the origins of the terms odd and even and related examples
y = sin x (odd) and y = cos x (even). After each part of this activity, students share their findings with the class. Consolidation could take a number of forms, such as this informal chart summarizing Part B:

	Even Degree	Odd Degree
Positive a	starts high, ends high	starts low, ends high
Negative a	starts low, ends low	starts high, ends low

It is important that the classification of functions as “odd’ and “even” (Parts C and D) is distinguished from functions of odd or even degree (Part B). Students can work in groups of four on Part E, presenting their graphs and explaining their findings to the class.

Student Activity

Part A – Shaping Up

Consider the following “families” of functions. [Suggested Window: , ]

Constant

(Degree 0)

Linear

(Degree 1)

Quadratic

(Degree 2)

Cubic

(Degree 3)

Quartic

(Degree 4)

y = 0 *

y = π

y = -3

y = -0.4

y = x *

y = 4x + 2

y = -2x – 3

y = -1.3x + 4.1

y = x² *

y = 0.2x² – x – 2

y = -3x² + 1

y = -2x² – x ! 1

y = x³ *

y = 3x³ – x – 1

y = -2x³ – 3x² – 5

y = -x³ + 2x

y = x⁴ *

x³ = 3x⁴ – 8x² – 1

x³ = -2x⁴ – 3x³ + 3x² + 1

x³ = -x⁴ + 3x² – x – 7

1. What do all of the graphs in each family have in common? Use graphing technology, if necessary, to visualize the graphs.

2. The first graph (*) is known as the most basic graph of each family, yet is not considered to be representative of the typical family member. Explain why.

3. How do the shapes of the graphs change as you increase the degree of the polynomial (i.e., as you move across the chart)? Be as specific as possible.

4. Predict the graph of a typical polynomial of degree 5. Graph the function y = x⁵ – 5x³ + 4x – 3 to confirm your prediction. How would the graph of y = -x⁵ + 5x³ – 4x + 3 differ?

5. Predict the graph of a typical polynomial of degree 6. Graph the function y = x⁶ – 5x⁴ + 6x² – 3 to confirm your prediction. How would the graph of y = -x⁶ + 5x⁴ – 6x² + 3 differ?

Part B – Who’s Got the Power?

Graph the sets of equations which have a common characteristic. Make a rough sketch of each set in your notes and label it appropriately, e.g., Set A, so that you can easily reference the graphs and simplify discussion after the activity. [Suggested Window: , ]

1. a) Graph the functions on a graphing calculator and determine what the graphs have in common.

SET A: y = 40x + 1, y = 3x³ – 2x² + 3x ! 7, y = 2x⁵ – 2x² – 3, y = 2x⁷ + 3x⁵ – 5x⁴ – 2x – 1

b) What characteristic of the equation causes this commonality? Explain why these properties exist.

c) Would the following functions share these characteristics? How would these graphs differ? Why?

SET B: y = -30x + 2, y = -2x³ + 2x² + 3x – 7, y = -x⁵ – 2x² + 3, y = -2x⁷ + 2x – 1

d) Summarize your findings for Set A by completing the following statement: “If a function has an odd degree and a leading coefficient that is positive, then the graph will…”

e) Write a similar statement for the graphs in Set B.

2. a) On a graphing calculator, graph the functions and determine what the graphs have in common.

SET C: y = 2x² + 1, y = 3x⁴ – 2x² + 3x – 7, y = 2x⁶ – 2x² – 3

b) What characteristic of the equation causes this commonality? Explain why these properties exist.

c) Would the following functions share these characteristics? How would these graphs differ? Why?

SET D: y = -2x² + 3x – 7, y = -x⁴ + 2x² + 3, y = -3x⁶ + 2x⁴ + 5x² – x – 1

d) Summarize your findings for Set C by completing the following statement: “If a function has an even degree and a leading coefficient that is positive, then the graph will…”

e) Write a similar statement for the graphs in Set D.

3. Is it fair to say that a person can tell the general shape of the graph of a polynomial function just by looking at one term in the equation? Explain.

Part C – Even in The Mirror…

When you first learned how to graph relations, you used tables of values to generate points and, in many cases, discovered patterns that expedited the process.

1. Fill in the tables of values, making note of any patterns you discover. In an effort to save trees, the column of x-values has only been written once. Since these calculations can be time-consuming, use technology to help [enter the equations using Y= and activate the TABLE feature].

x	Y₁ *= x*²	Y₂ *= x*⁴	Y₃ *= x*⁶	Y₄ *= x*⁴ + 3x²	Y₅ = 2x⁶ *– x*² + 1	Y₆ *= x*⁶ + 2x – 3
-3
-2
-1
0
1
2
3

2. All but one of the functions in #1 is an even function. Which one is not an even function? Why?

3. Describe the type of symmetry that is demonstrated by the graphs of even functions. (You can graph the functions using technology if you need some help).

4. Write a formal definition for an even function.

5. Is it reasonable to say that even functions demonstrate the property that f(x) = f(-x) for all x-values? Explain, using the function f(x) = 2x⁶ – x² + 1 as an example.

6. How can you tell by looking at the equation of a polynomial whether it is an even function or not?

Part D – Odd Origins…

1. Fill in the tables of values, making note of any patterns. Use technology.

x	Y₁ *= x*	Y₂ *= x*³	Y₃ *= x*⁵	Y₄ *= 2x³ + x*	Y₅ = 2x⁵ *– x*³ + 1	Y₆ =!x⁵ *+ x*³ – 2x
-3
-2
-1
0
1
2
3

2. All but one of the functions above is an odd function. Which one is not an odd function? Why?

3. Write a formal definition for an odd function.

4. Graph each odd function. Describe the type of symmetry that is demonstrated by the graphs of odd functions. [Hint: Consider joining any two of these symmetric points.]

5. Is it reasonable to say that odd functions demonstrate the property that f(-x) = -f(x) for all x-values? Explain, using the function f(x) = -x⁵ + x³ – 2x as an example.

6. How can you tell by looking at the equation of a polynomial whether it is an odd function or not?

Extensions

1. Choose any quartic function and graph it using technology, e.g., y = x⁴ – 3x³ – 5x² + 3x – 1. Change the value of the constant term and examine the new graph. Try a wide range of values (large, small, positive, negative, etc.). What effect(s) does changing the value of the constant have on the graph? Is this consistent with what you know of linear and quadratic functions? Is this consistent with polynomials of any degree? Is it consistent with functions of any kind? Explore with technology to verify your answer. [Suggested Window (also for #2): , ]

2. Choose any cubic function (e.g., y = x³ – 3x² – 2x + 3) and graph it using technology. Change the leading coefficient and examine the new graph. Try a wide range of values (large, small, positive, negative, etc.). What effect(s) does changing the value of a have on the graph? Is this consistent with what you know of linear and quadratic functions? Explain. Is this consistent with polynomials of any degree? Explore with technology to verify your answer.

3. Using a graphing calculator, students investigate how changing the other coefficients in a polynomial function affect the graph. For example, start with function f(x) = ax⁶ + bx⁵ + cx⁴ + dx³ + ex² + fx + g, and describe how the graph changes as you vary each of the coefficients. Some computer software allows students to animate the graph as the coefficient changes. If such software is not available, a spreadsheet is also effective in comparing graphs with various coefficients. If time allows, the game Target can be played, requiring students to adjust the coefficients of a polynomial in order to hit as many points as possible. Alternately, the teacher could provide students with a number of points on the graph of a polynomial equation and students could submit an equation obtained by trial and error on a graphing calculator or Ministry-licensed software, such as Zap-a-Graph.

4. You have learned how to identify symmetry about the y-axis. Why has symmetry about the x-axis been ignored? (Why can’t functions have symmetry about the x-axis?)

Follow-up Questions

1. Given several different types of polynomial functions (the graph, the equation, or the table), identify the type of function (odd, even, or neither).

2. Match an equation with a rough sketch of a graph (only end behaviour and symmetry are evident).

Assessment & Evaluation of Student Achievement

· Communication – Formative assessment of student presentations or journal entries summarizing their findings for some or all parts of the activity.

· Knowledge/Understanding – Formative assessment through questioning or a warm-up the next day, e.g., classify functions as odd or even degree.

· Thinking/Inquiry/Problem Solving – Ask questions, such as “Do all functions of odd degree have at least one x-intercept? Why?” or “Why does adding a constant to an odd function ruin the symmetry?” or “If is a point on an odd function, then is on the function as well? Explain.” These questions could be posed on a quiz, informally as discussion items, or on an in-class assignment.

Accommodations

· Students who are skilled with graphing calculators can be paired up with those experiencing difficulty.

· Graphs of the most common polynomial functions could be displayed as posters around the classroom.

· The teacher may want to break up the activity into distinct parts (possibly stations) and bring closure to each part by discussing (or presenting) the answers so that students can proceed with confidence.

Resources

Target graphing calculator program. Available from – www.ti.com.

Activity 1.3: Throwing Interceptions

Time: 1.25 hours

Description

Students examine various polynomial functions and their graphs. They try to develop relationships between the nature of zeros of the polynomial functions and the corresponding graphs. Students then sketch graphs of polynomial functions given in factored form.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision making, problem-solving, time, and resource management skills;

CGE7j - a responsible citizen who contributes to the common good.

Strand(s): Advanced Functions

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees.

Specific Expectations

AF1.04 - sketch the graph of a polynomial function whose equation is given in factored form.

Prior Knowledge & Skills

· the ability to interpret the graphical significance of single and double roots of quadratic equations

· proficiency with a graphing calculator or graphing software

Planning Notes

· Students need access to a graphing calculator or graphing software, as well as graph paper.

Teaching/Learning Strategies

Teacher Facilitation

The teacher reviews terminology, such as degree of a function, zeros, standard form, and factored form. The teacher introduces the term turning point as a local minimum or maximum value of the function. When students have completed questions #2 and #6, they could share their conclusions with the class. Determining the number of turning points in #6 will be challenging for most students [(# of turning points) = (# of zeros) + (# factors that occur an even number of times) – 1]. Teacher facilitation may be necessary to help students distinguish single zeros that just pass through the x-axis, double zeros that “bounce” off the x-axis, and triple zeros which “inflect” on the axis. An informal discussion extending this idea to quadruple or quintuple zeros might be interesting for students (the idea that there was a change in direction, but then a simultaneous change back, several times). Students should understand that zeros occurring an odd number of times pass through the x-axis and zeros occurring an even number of times bounce off. In questions #9 and #10, the teacher may need to draw out the idea that the sketches are just sketches. The exact location of the local minimum and maximum values of the function cannot be easily ascertained. The functions in #9 all have a leading coefficient of 1. By considering other values of a, graphs may be vertically stretched, compressed, or reflected in the x-axis. The teacher can draw out the connection to end behaviours in the previous activity and families of functions in the next activity.

Student Activity

Examine the graphs of polynomial functions with equations given in factored form and try to develop a relationship between the factors of a polynomial function and characteristics of the corresponding graph.

1. Use a graphing calculator or graphing software to create sketches of the following functions and complete the table. Label your graphs so that you can refer to them later on.

Function	Degree of Function	# of Zeros/ x-intercepts	# of Turning Points
f(x) = (x – 2)
f(x) = (x – 2)(x + 1)
f(x) = (x – 2)(x + 1)(x + 3)
f(x) = (x – 2)(x + 1)(x + 3)(x)
f(x) = (x – 2)(x + 1)(x + 3)(x)(x – 1)

2. Based only on the findings in the table, write a conclusion relating the degree of a function, the number of zeros/x-intercepts, and the number of turning points.

3. Complete the statement: If a polynomial of degree n has ________ real and distinct zeros, then the graph will have ________ turning points.

Unfortunately, not all polynomials behave in this manner. All of the functions in the previous question were “nice” polynomials in that no factors occurred more than once and all of the factors were linear (i.e., of the form x – a); the number of x-intercepts matched the degree of the polynomial. For example, recall the following quadratic functions and their graphs from previous grades:

f(x) = x² ! 4

f(x) = x²

f(x) = x² + 4

It should be clear that the function on the left is “nice” by our definition since it is degree 2, has two real and distinct zeros, and has one turning point. The second function is not nice because it has repeated zeros; the third function is not nice because it does not have any real zeros at all. This activity only considers the first problem (having repeated zeros like f(x) = x²)

4. The following table illustrates how the features of a graph are affected by factors that occur more than once in a polynomial equation. Use graphing technology and graph one function at a time.

Function	Degree of Function	# of Zeros/ x-intercepts	# of Even Zeros	# of Turning Points
f(x) = (x – 2)²
f(x) = (x – 2)³
f(x) = (x – 2)⁴
f(x) = (x – 2)²(x + 1)
f(x) = (x – 2)²(x + 1)²
f(x) = (x – 2)³(x + 1)²
f(x) = (x – 2)³(x + 1)²(x)

(# of Even Zeros refers to the number of zeros that occur an even number of times)

5. Discuss the characteristics of the graph that arise when a function has a factor that appears twice, three times, four times, etc. Provide examples and sketches to illustrate.

6. Does your conclusion from question #2 still hold true? If not, modify it so that it does, i.e., How can you determine the number of turning points just by looking at the equation?
[Hint: Consider the only function for which the old rule works. How is it different from the others?]
Answer: (# of turning points) = (# of zeros) + (# factors that occur an even number of times) –1.

7. a) Suppose you were given the sketch of a polynomial function and asked how many times a certain factor occurred in the corresponding equation. Do you think it would be possible to determine with certainty whether a factor occurred 3, 5, 7, or another odd number of times? Explain.

b) Why was a factor occurring only once not included in this discussion?

c) How would the graphs of f(x) = (x – 3)(x + 2)(x – 1) and f(x) = (x – 3)(x + 2)³(x – 1) differ?

8. Similarly, do you think it would be possible to distinguish whether a factor occurred 2, 4, or 6 times? Explain.

9. By considering the number and type of zeros, sketch the following functions without the use of graphing technology. Write the corresponding equation on the graphs.

a) f(x) = (x + 2)(x + 1)(x)	b) f(x) = (x – 2)(x – 3)(x + 4)²	c) f(x) = (x + 2)²(x – 4)²
d) f(x) = (x + 5)³(x – 5)(x)	e) f(x) = (x – 1)³(x + 4)²(x + 2)	f) f(x) = (x + 6)⁴(x + 1)(x³)
g) f(x) = (x – 2)(x + 1)²(x⁴)	h) f(x) = (x + 3)(x – 4)³(x + 2)(x²)	i) f(x) = (x + 5)⁵(x + 1)⁵(x – 3)⁵

10. a) Can two people sketching the same graph have higher or lower hills and valleys? Why?

b) How could you get a more accurate idea of where these turning points are located? (Later in the course, you will learn powerful methods to find the exact location of these important points.)

Follow-up Questions

Assign questions similar to question #9 and others with leading coefficients not equal to 1, such as
f(x) = -2(x + 2)²(x + 1). Present students with a question that requires them to match function equations with the corresponding graphs, which are drawn by hand or generated by graphing technology. There is an opportunity to include some graphs with missing information requiring students to use some reasoning skills and setting the stage for future activities. Extending examples include:

1. Sketch the graph of f(x) = (x – 1)(x – 2)²…(x – 9)⁹(x – 10)¹⁰.

2. Would the graph of f(x) = (x – 1)(x – 2)²(x – 3)³ look different from the graph of
f(x) = (x – 1) (x + 2)²(x – 3)³? Explain, citing specific differences and similarities.

Extensions

· Ask students to describe how to classify equations given in factored form as odd or even functions.

· Students explain why zeros that occur an odd number of times correspond to the graph passing through the x-axis and zeros that occur an even number of times correspond to bouncing off the
x-axis.

· Ask students how they would graph a function like y = (x² – 4) (x² – 9) and if functions like
y = (x² + 4)( x² + 9) are easier to graph or more difficult. Why?

Assessment & Evaluation of Student Achievement

· Knowledge/Understanding – Complete a chart, e.g., question #1 and #4, without using technology.

· Application – Given the equation of a function, sketch the graph, as in question #9.

· Communication – Ask students to explain the relationship between the nature of zeros of polynomial functions and the corresponding graphs and to provide examples.

Accommodations

· Students who are competent with the graphing technology could be designated as “experts” and provide assistance to other students. Alternately, students could be grouped in weak/strong pairs as it pertains to technology or based on skill level in constructing graphs by hand.

Activity 1.4: We Are Family!

Time: 1.25 hours

Description

Given the real and complex zeros of a polynomial function, students determine the general equation of the polynomial (in factored form) and sketch the corresponding graph. Students also determine the equation of the specific function in a family of curves that passes through a given point.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written materials effectively.

Strand(s): Advanced Functions

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees;

AFV.02 - demonstrate facility in the algebraic manipulation of polynomials.

Specific Expectations

AF1.04 - sketch the graph of a polynomial function whose equation is given in factored form;

AF2.05 - write the equation of a family of polynomial functions, given the real or complex zeros [e.g., a polynomial function having non-repeated zeros 5, -3, and -2 will be defined by the equation
f(x) = k(x – 5)(x + 3)(x + 2) for ].

Prior Knowledge & Skills

· familiarity with the sets of numbers (integer, rational, irrational, complex)

Planning Notes

· Students should have an ample supply of graph paper.

· The teacher may want to have graphing technology available for students to check their work.

Teaching/Learning Strategies

Teacher Facilitation

The teacher reviews the definition of a zero of a function and the concept of expanding, primarily as it pertains to binomials. Students should be comfortable expanding binomials containing irrational and complex numbers (e.g., (x + 2i)(x – 2i) = x² + 4). Since performing operations with complex numbers is an expectation only in the MCR3U course, the teacher should do a few straightforward examples prior to beginning the activity. It should be noted that the idea of zeros is usually reserved for dealing with functions that represent a real-life application and so the complex zeros should not be emphasized (although there are applications in science that do require the use of complex numbers). Students should be aware that if they are not given ALL of the zeros, then there are many possibilities for the equation of the function. In this activity, however, students can assume that all zeros are given. Also, when sketching functions in general, students need to realize that without knowing the k-value, the graph may not be stretched or compressed appropriately and, more importantly, it may be upside down (reflected in the x-axis). Depending on timing, Part B might need to be assigned as homework. Part A can certainly be condensed to include only a few examples of each type. After Part A, students should be able to complete Part B, #1.

Teacher Notes

In this activity, only polynomials containing rational coefficients are considered, which is why irrational and complex zeros are given in pairs. The teacher may opt to use questions #7 and #9, involving conjugates, as extensions. To explain the meaning of complex zeros graphically, it may be sufficient to consider the equation x² + 1 = 0 and relate the fact that is has no real solution to the fact that the graph
y = x² + 1 has no x-intercepts (since the parabola opens up and the vertex is above the x-axis). So, although the function f(x) = x² + 1 does indeed have zeros at i and -i, these zeros do not correspond to
x-intercepts. This idea can be applied whenever a factor yielding complex roots (e.g., x² + 1) occurs in the equation of the polynomial. Emphasis is placed on the connection between the graph and the equation and NOT on the algebraic manipulation that comes in Unit 3. Students are not even asked to expand the general equations fully, only enough to eliminate irrational and complex values from the equation.

Wherever possible, the teacher should try to incorporate context in additional examples and homework questions and/or extensions. A variety of these types of questions should be available in textbooks.

The teacher might be tempted to teach sum and product of the roots for polynomials, but this idea may be better placed in Unit 3 with the Algebraic Manipulation.

Student Activity

Part A

In the following activity, you will be asked to find the general equation of a polynomial function given the zeros of the function. Consider the following example:

Example: A polynomial function has zeros at 5 and -7. Find the general equation of this family of polynomials, and sketch the graph of the polynomial.

Solution:

The required equation will take the form f(x) = k(x – 5)(x + 7), since substituting either 5 or –7 into the equation should yield a result of 0; there should be factors of (x – 5) and (x + 7) in the equation. The unknown k is simply a constant that cannot be determined at this point.

1. Is the graph the only (correct) answer? Explain. In the example, the right-hand side of the equation could be expanded to f(x) = k(x² + 2x – 35), but since that form has little value to us, we will leave the equations for the following questions in factored form, as long as all values in each factor are integers.

2. The zeros of a polynomial function are given. Determine the general equation of the polynomial function and sketch the corresponding graph on a separate sheet of paper. Be sure to label the graphs with the equation. The first equation is done for you.

Zeros	General Equation of the Family
a) -3, -2	f(x) = k(x + 3)(x + 2)
b) 0, 1, 2
c) -7, 3, 5, 4
d) 1, 2, 3, 4, 5

3. a) Why do we need to put a k-value in the equations?

b) Should a scale be drawn on the y-axis? Explain.

c) How does varying k affect the shape of the graph? (What transformation(s) does k cause?)

d) How does varying k affect the zeros of the graph?

4. Sketch the following functions, containing repeated zeros, using the knowledge you have learned.

Zeros	General Equation of the Family
a) 4, 4	f(x) = k(x – 4)²
b) 0, -3, -3, -3
c) 0, 7, 7, 4, 4, -5
d) 1, 2, 3, 4, 5

5. In reality, the zeros are rarely nice integer values, but even if the zeros are non-integer values, we can determine the equation for the function. Determine the general equation of the following functions.

Zeros	General Equation of the Family
a)	f(x) = k(2x – 1)(3x + 2)(x – 4)
b)
c)
d) 0.1, 0.2, 0.3, 0.4
e)

6. There may also be some irrational zeros that need to be accounted for. Simply expand factors containing irrational values so that the equation contains only rational numbers.

Zeros	General Equation of the Family
a)	or
b)
c)
d)

7. Why do you suppose you were only given functions with zeros that were conjugate pairs, like and ? [Hint: Try to write the equation of a polynomial with zeros at and .]

8. There may be complex zeros that need to be accounted for. They can be handled the same way as irrational zeros. Just do some expanding and remember that complex roots do not correspond to x-intercepts on the graph, so you will have to use your reasoning skills. The graph of some of these functions may have more than one possibility. Also, you may not be able to determine exactly how high or low a turning point goes. You can substitute appropriate x-values into the equation to get a rough idea of where the turning points are. In future units, you will learn powerful ways of locating turning points exactly.

Zeros	General Equation of the Family
a) i, -i	f(x) = k(x – i)(x + i) or f(x) = k(x² + 1)
b) 3, 1 + i, 1 – i
c) 0, 2 + 3i, 2 – 3i, 4, -5
d)

9. Why do you suppose the complex zeros in the previous question occurred in complex conjugate pairs? (in the form a + bi and a – bi). [Hint: Consider an example where the given zeros are
3 + i, 2 – i). Could you still get the general equation of the family? Explain why or why not.]

Part B

In Part A, we determined the general equation for a family of curves given the corresponding zeros. To find one specific equation (one specific member of the family), you would need to know one more piece of information, such as a point on the function. Since a point on a curve must satisfy the equation of the curve, the specific value of k can be determined quite easily.

Example: Find the equation of the polynomial with zeros of 1, 2, and 3 that passes through (4, 12).

Solution: The family of equations will have the form y = k(x – 1)(x – 2)(x – 3). Since (4, 12) is on the curve, it must satisfy the equation. 12 = k(4 – 1)(4 – 2)(4 – 3) yields 12 = 6k (k = 2). The equation of the required polynomial is y = 2(x – 1)(x – 2)(x – 3).

1. The zeros and a point on a polynomial function are given. Determine the equation of the function.

Zeros	Point on the Function	Zeros	Point on the Function
a) 1, 1, 2	(3, -1)	e) 2, 2, 2, 2, 2	(4, 32)
b)	(2, 5)	f)	(2, 2)
c) 2 – 3i, 2 + 3i, 1, 2	(7, -7)	g)	(2, 13)
d)	(-2, -4)	h)	(4, 0)

2. Why were you unable to determine the value of k for h)? Explain using an algebraic argument and using a graphical argument.

Follow-up Questions

Additional questions like #1 from Part B and other examples embedded in a context, if available.

Consider the following example:

The graph shown was produced using a motion sensor which measured the height of a ball as it bounced on the floor. The graph shows the times when the ball hit the ground for one particular bounce, and the height obtained for three successive bounces. Based on this information, determine the equations of the parabolas representing these three bounces. (Hint: The k value can be determined from the 2^nd bounce and will be the same for each parabola.)

Extension

What would happen if a function had an odd number of irrational or complex roots? Be specific.

Could the equations in question #5 be written in fraction form instead, i.e., instead of f(x) = k(2x – 1)(3x + 2)(x – 4)? Explain.

Assessment & Evaluation of Student Achievement

· Knowledge/Understanding – Given the zeros of a function, students determine the general equation.

· Application – Given the zeros and a point on a function, students determine the equation and sketch the graph.

· Communication – Part A, questions #3 and #9, and Part B, #2. An assignment or group presentation could be assigned in which students are given some information about a mystery function (the zeros and a point on the function or the graph of the function) and work in groups to determine the unknown or alternate form for the function. Students could present their solutions and justify their reasoning for determining the equation and/or graph.

· Thinking/Inquiry/Problem Solving – Given all but one of the zeros and one or more points, determine possible equations or graphs for functions.

Accommodations

The teacher may consolidate this activity by taking up examples of each question in Part A and question #1 in Part B. A simplified version of this activity containing only questions like Part B, #1 (with examples) may be more accessible to some students but still effective as a learning tool.

Activity 1.5: Equation Station

Time: 2.5 hours

Description

Students determine the equation of a graph of a polynomial using appropriate methods, such as using the zeros, trial and error with a graphing calculator, and finite differences in conjunction with reasoning skills.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written materials effectively;

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4b - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability.

Strand(s): Investigating the Graphs of Polynomial Functions, Manipulating Algebraic Expressions

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees;

AFV.02 - demonstrate facility in the algebraic manipulation of polynomials.

Specific Expectations

AF1.05 - determine an equation to represent a given graph of a polynomial function, using methods appropriate to the situation (e.g., using the zeros of the function; using a trial-and-error process on a graphing calculator or graphing software; using finite differences);

Prior Knowledge & Skills

· graphing technology features, such as zooming in on zeros

Planning Notes

· Students need access to a graphing calculator or graphing software and spreadsheet software.

Teaching/Learning Strategies

Teacher Facilitation

Some parts of this activity are better suited to a teacher-directed approach, e.g., the first section of
Part A; other sections could be assigned to students in pairs. Students should be encouraged to use knowledge from previous courses where appropriate – especially as it pertains to linear and quadratic functions, e.g., if the function is quadratic, students may use the form y = a(x – h)² + k to quickly determine the equation.

To complete Part C as intended, students need to complete Activity 1.1 or equivalent work with a spreadsheet. The equations (answers) corresponding to the graphs are provided.

Teacher Notes

Part A revisits similar concepts to Activity 1.4, but leads into the more realistic scenarios where only some of the zeros are known. It is important to note that when students see a graph or part of a graph, it is not always obvious what type of function is being displayed (a graph might appear to be quadratic, but it may actually be part of a cubic or quartic graph). Students should realize that if you only know some of the zeros, determining the equation becomes much more difficult and more advanced strategies need to be implemented. Even if a function has a zero at , for example, it will be difficult to ascertain that exactly using a graphing calculator. More sophisticated strategies and higher-order thinking are required. There are many methods for determining the equation of a polynomial function; a few methods are explored. For example, given a number of points on the graph of a function, a system of n equations and n unknowns can be obtained quickly by substitution, but solving the system can get tedious and time consuming. Since the focus of this unit is more graphical than algebraic, this method is not examined.

In Part B, when students are finding the unknown factors of the equation, they are setting up the ideas of factoring using long division in Unit 3. If students can determine other factors in an equation using this type of argument, it would be extremely valuable to share these findings with the class. However, the emphasis in this unit is not on the algebraic manipulation. The hope is that the extensive exploration with the graphical models will bring great understanding when the algebra is explored fully in later units.

Student Activity

Part A – Using All Zeros

You have learned that finding the roots of an equation is equivalent to finding the x-intercepts of a corresponding graph and if you know the x-intercepts of a function, you can get a possible equation for the polynomial. Consider the following example:

Determine the equation of the function shown on the graph.

Solution: Since the function has x-intercepts (zeros) of 2 and 5, the equation must have factors of (x – 2) and (x – 5), so the equation will take the form f(x) = k(x – 2)(x – 5). Since the point (1, 2) is on the curve, it must satisfy the equation. 2 = k(1 – 2)(1 – 5) yields 2 = 4k, so . The required equation is .

Note: We have assumed that the graph shown is a quadratic function. Is it possible that the graph shown is part of a cubic or quartic graph? How can you verify your hypothesis?

Using the same method, find the equation of the polynomials represented by the following graphs. You can assume that all of the important features of each function are evident, i.e., there are no additional turning points or x-intercepts, etc. In the previous example (and in question #1), the degree of the polynomial matches the number of x-intercepts, i.e., the function has no repeated zeros and no complex zeros. If there are repeated zeros, such as the graphs in question #2, these can be dealt with by using factors more than once in the equation, as seen in previous work. For example, the equation in question
2 a) will have a factor of (x – 2)³. Can you explain why?

1. a) 2. a)

b) b)

c) c)

Part B – Using Translations!

Consider the following example:

The graph shown to the right appears to be a cubic with only one real zero and it is not at an obvious (integer) point. If you examine the graph carefully, though, it clearly passes through the points (-3, -3), (1, -3), (3, -3), and (-1, -11). If we consider translating this function up 3 units, the equation would have zeros at x = -3, x = 1, and x = 3 (because of the first 3 points) and would have an equation of the form f(x) = k(x + 3)(x – 1)(x – 3).

Since (-1, -11) was on the original function, (-1, -8) must be on the translated function and will therefore satisfy its equation. Substituting (-1, -8) into the equation yields a k-value of , so the translated equation is and the equation of the graph shown is . The right side of this equation could be expanded to put the equation into standard form.

1. Using the same method, find the equation of the polynomials represented by the following graphs. Assume that all of the important features of each function are evident, i.e., there are no additional zeros, turning points, etc.

a) b)

Extensions

Extension A – Complex Zeros

If a polynomial has complex zeros, e.g., a quartic with only 2 real zeros, it may be necessary to create a quadratic factor of the form ax² + bx + c and then determine the values of a, b, and c by substituting points that are on the graph. In some cases, algebra and problem solving are necessary to determine the equation of a polynomial function exactly. By using trial and error and a graphing calculator, you may be able to get a fairly close approximation of the equation. Make sure that you use all of the known information to narrow down your exploration.

Consider the following example:

Determine the equation of the function shown on the graph.

Solution: The function appears to be a quartic, with zeros at x = 2 and x = 3, so the equation will have the form f(x) = (x – 2)(x – 3)(ax² + bx + c). Since the function clearly passes through the point (0, 6), those coordinates satisfy the equation, yielding c = 1. Using trial and error, a graphing calculator, or other problem-solving methods, the values of a and b can be obtained (a = 1 and b = 0).

1. Determine the equations of the following functions:

a) b)

2. Describe some of the limitations that you discovered while using this method.

Extension B - Finite Differences Revisited!

As seen previously, finite differences can be helpful in determining the equation of a polynomial function.

Consider a polynomial graph passing through the points (1, 7), (2, 22), (3, 45), (4, 76), and (5, 115).

We could make an educated guess about the type of function, substitute the points into the equation, and then solve a system of equations. Alternately, we could calculate the finite differences for the function.

Since the second differences are constant, it means that the function is quadratic (of the form
f(x) = ax² + bx + c) Also, the second differences are equal to 2a, which implies that a = 4. Therefore, the function is f(x) = 4x² + bx + c. By backing up the finite differences, you can obtain the point (0, 0), implying that c = 0. Now there are several options available to determine that b = 3, and the required function is f(x) = 4x² + 3x.

1. Use this method to determine the equations of the following polynomial functions. In previous exercises, we determined the equation in factored form. It may be easier to determine the equations of these functions in standard form.

a) b)

Follow-up Questions

The teacher provides more examples (from Part A especially). In Extension B, it may be helpful to draw very rough sketches of graphs and indicate the points they go through. Graphs drawn to scale can be very difficult to generate and/or work with, and it is important for students to use their judgement when considering how reasonable a graph is. If time allows, the game Target could be played on graphing calculators, requiring students to adjust the coefficients of a polynomial in order to hit as many points as possible. A teacher could easily improvise such a game by providing a number of points and have students submit an equation obtained by trial and error on a graphing calculator or Zap-a-Graph.

Assessment & Evaluation of Student Achievement

· Application – Given the graph of a function, students determine the equation, as in Part A, #1 and #2. This could be collected as formative assessment for this concept.

· Thinking/Inquiry/Problem Solving – Students pull a graph out of a hat and determine the equation of the function as accurately as possible. Graphs used for this task should not be straightforward graphs, such as those from Part A, but the more complex types from the remainder of the activity. If trial and error is needed for the graph involved, the activity could take the form of a contest to see how closely student equations fit a curve. The teacher can demonstrate an appropriate regression and calculate correlation values to determine a winner of the contest.

· Communication – The proper use of notation, mathematical form, and conventions could be assessed throughout the activity. Some of the questions could be handed in and assessed based on the quality of the communication in determining and justifying the equations.

Answers

Part A

1a) y = (x + 1)(x – 2)(x – 7)	b) y =-2(x + 2)(x – 4)	c) y = 0.25(x + 4)(x + 2)(x – 1)(x – 3)
2a) y = x(x – 2)³	b) y = -0.25(x + 2)²(x – 3)²	c) y = 0.25x(x + 3)²(x – 2)(x – 3)

Part B

1. a) y = 0.125(x – 2)(x + 3)(x – 7) + 1 b) y = x⁴ – 4x² + 5 or y = x²(x – 2)(x + 2) + 5

Extension A

1. a) y = 2x³ – x + 1 or y = (x + 1)(2x² – 2x + 1) b) y = (x – 2)²(x² + x + 1)

Extension B

1. a) y = -2x² + 12x + 9 b) y = x³ – 2x² – 2x – 1

Accommodations

· Individual instruction could be provided to students having difficulties while students demonstrating aptitude work independently on extending questions presented in the activity.

· Less wordy examples could be provided for students with reading or language difficulties.

Resources

Target graphing calculator program. Available at – www.ti.com.

Graphmatica graphing software. Available at – www.graphmatica.com.

Winplot graphing software. Available at – http://math.exeter.edu/rparris/winplot.html.

Activity 1.6: Get Back To Your Roots

Time: 1.25 hours

Description

Students determine the roots of polynomial equations by examining the x-intercepts of the corresponding graphs, using technology where appropriate.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4b - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability.

Strand(s): Investigating the Graphs of Polynomial Functions, Manipulating Algebraic Expressions

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees;

AFV.02 - demonstrate facility in the algebraic manipulation of polynomials.

Specific Expectations

AF1.01 - determine, through investigation, using graphing calculators or graphing software, various properties of the graphs of polynomial functions (e.g., determine the effect of the degree of a polynomial function on the shape of its graph; the effect of varying the coefficients in the polynomial function; the type and the number of x-intercepts; the behaviour near the x-intercepts; the end behaviours; the existence of symmetry);

AF2.04 - determine the real roots of non-factorable polynomial equations by interpreting the graphs of the corresponding functions, using graphing calculators, or graphing software.

Prior Knowledge & Skills

· use of a graphing calculator and/or spreadsheet software

Planning Notes

· Students need access to a graphing calculator or spreadsheet software.

Teaching/Learning Strategies

Teacher Facilitation

The teacher should distinguish between the terms roots (of an equation), zeros (of a function) and
x-intercepts (of a graph). The teacher also emphasizes the connection between finding the roots of an equation and finding the x-intercepts of the corresponding graph, especially the idea of setting y = 0 and solving for x. In this activity, students are essentially dealing with the nicest form of polynomial equations (already factored) and the ugliest form (non-factorable). Although some of the equations can be factored (question #3), this idea should not be emphasized since students will learn additional factoring methods in Unit 3. It is important that students realize when graphing technology is an appropriate tool (not in question #1) and that polynomials given or obtained in standard form may or may not be factorable. In fact, when they graph these polynomials using technology, they should see that the x-intercepts are at “nice” locations and, based on previous work, they should be able to come up with the factored form of the equation. Some of these polynomials have been included to help promote discussion.

Teacher Notes

When asked to solve equations in this activity, it is assumed that only real roots are considered.

Student Activity

To find the x-intercept of any graph, you can set y = 0 in the equation and solve for x. In Grade 9, you found the x-intercept of a linear function like y = 3x – 4 by setting y = 0 and solving 0 = 3x – 4.

Similarly, in Grade 10, you found the x-intercept(s) of a function like y = 3x² – 4x + 5 by setting y = 0 and solving 0 = 3x² – 4x +5. The act of solving an equation is exactly equivalent to finding the x-intercepts of the corresponding graph or, put differently, finding the zeroes of the function. Just like some quadratics cannot be factored (the x-intercepts were at places like or there were no x-intercepts at all), polynomials of higher degree cause similar dilemmas. In fact, it is impossible, or at least extremely difficult, to find the exact value of the zeros of many polynomials. Graphing technology is extremely effective in obtaining approximate values for the zeros of these polynomials. It is important to note however, that although the zooming features of graphing technology can give fairly accurate values for the x-intercepts of graphs, they are generally just approximations of the actual values. For example, if an x-intercept appears to be extremely close to 0.5, it might be reasonable to assume that the intercept is. Consider x-intercepts of or which are virtually indiscernible using technology. In other words, for applications that require exact values, graphing technology has limitations and more sophisticated algebraic skills are required. Some of these skills are introduced in Unit 3. [Note: The quadratic formula can determine the exact value of zeros for polynomials of degree 2, and there is a more complex formula to help with cubic polynomials, but higher degree polynomials are much more difficult.]

1. State the x-intercepts of the following functions:

a) y = (x – 2)(x – 6)	b) y = 3(x – 3)(x + 2)(x – 1)
c) y = (x – 2)²(x + 1)³	d) y = 4x⁵(2 – x²)

2. Why was a graphing calculator not required for question #1?

3. Determine the x-intercepts of the following functions by factoring first:

a) y = x³ – 5x	b) y = x² – 3x – 10
c) y = 3x² – 6x + 3	d) y = x⁴ – x² – 6

4. Is it always possible to factor the polynomial to determine the x-intercepts? Explain.

5. Is it possible that a function has no x-intercepts? Explain, citing an example.

6. Solve each equation by:

i) Writing the equation of the corresponding graph (you may need to rearrange the equation first);

ii) Using a graphing calculator to find the x-intercepts of the corresponding graph;

iii) Factoring in order to state the roots of the equation.

[Use the zoom or zeros features to get an estimate that is accurate to 3 decimal places.]

a) 0 = 6x² – 5x – 4	b) 0 = 3x³ – 5x² + 1	c) 0 = x³ –3x² – x + 3
d) 3 = x² – 6x⁵	e) 4 = 7x² – x	f) 3x⁵ = x⁶ + 1
g) 0 = x⁴ + 10x³ + 35x² + 50x + 24			h) x⁴ – x = 1

7. Which equations from question #6 could be re-written in factored form? (Hint: there are three equations.) Write the equations in factored form and use the calculator to ensure that they are correct.

8. Is it possible that equations such as these have no solutions? Explain, using a graphical argument.

9. Is it possible that a polynomial equation of odd degree has no solutions? Explain.

10. How many x-intercepts can a function of degree 5 have? degree 6? Explain.

11. Solve the following equations for , if possible:

a) 0 = x² + 1	b) 0 = 3x⁴ + 4x² + 1	c) 0 = 2x³ – 7x + 2
d) 0 = x⁴ – x² – 1	e) 0 = 6x⁵ + 5x – 4	f) 0 = x⁶ + 2x² – 3
g) 0 = x⁴ + 8x³ + 15x² + 10x + 2			h) 0 = 3x⁵ – 4x + 1

Follow-up Questions

Assign some additional equations like #11, but also have students find the x-intercepts for non-factorable functions. Also have students solve some ugly equations, such as . There are no easy ways to solve such equations, so obtaining an estimate using graphing technology will have to suffice.

Extensions

1. Have students solve scrambled equations, such as 4 – x² = x³ – x, and then find the point(s) of intersection of two functions. Students could graph the functions and the function of intersection and see what they notice, e.g., y = x³ – x and y = 4 – x². If you equate them, you get 0 = x³ + x² – x – 4; if you graph the first two equations, along with y = x³ + x² – x – 4, the x-intercepts of the new graph correspond to the points of intersection of the original curves. The teacher could have students explain why this is this case.

2. Solve contextual problems, such as “The height in metres of a soccer ball t seconds after being kicked is given by h = -5t² + 20t. For how long is the ball higher than 11 m?” Since quadratics are done in previous grades, an attempt should be made to incorporate contexts involving polynomials of higher degree.

Assessment & Evaluation of Student Achievement

· Knowledge/Understanding - A pencil-and-paper quiz requiring students to state the roots of polynomial equations that are factored, or x- intercepts of polynomial graphs, or easily factorable and a quiz allowing graphing technology could be provided for non-factorable equations.

· Communication – Present questions such as, Explain why finding the x-intercepts of a graph is equivalent to solving the corresponding equation, or How can you tell by looking at the equation
y = 3x² – 2x – 5 that the graph will have two x-intercepts?

· Thinking/Inquiry/Problem Solving – Have students answer questions like #8-10 or others involving an exploration of the nature of roots/number of x-intercepts of a given polynomial function.

Accommodations

· Students with strong graphing calculator skills could be paired with students experiencing difficulties. Posters or reference sheets outlining specific keystrokes for the graphing calculators could be made available in the classroom.

Activity 1.7: Game, Set, Match – Summative Assessment Activity

Time: 2.5 hours

Description

Students demonstrate knowledge of polynomial functions and their graphs. In Part A, students complete a paper-and-pencil test focusing on the critical skills learned in the unit. In Part B, students use problem-solving techniques to match polynomial functions given in graphical, algebraic, and numeric forms and determine the missing forms for those that do not have a match.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE4a - a self-directed, responsible, life long learner who demonstrates a confident and positive sense of self and respect for the dignity and welfare of others.

Strand(s): Advanced Functions

Overall Expectations

AFV.01 - determine through investigation, the characteristics of the graphs of polynomial functions of various degrees;

AFV.02 - demonstrate facility in the algebraic manipulation of polynomials.

Specific Expectations

AF1.01 - determine, through investigation, using graphing calculators or graphing software, various properties of the graphs of polynomial functions (e.g., determine the effect of the degree of a polynomial function on the shape of its graph; the effect of varying the coefficients in the polynomial function; the type and the number of x-intercepts; the behaviour near the x-intercepts; the end behaviours; the existence of symmetry);

AF1.02 - describe the nature of change in polynomial functions of degree greater than two, using finite differences in tables of values;

AF1.03 - compare the nature of change observed in polynomial functions of higher degree with that observed in linear and quadratic functions;

AF1.04 - sketch the graph of a polynomial function whose equation is given in factored form;

AF2.04 - determine the real roots of non-factorable polynomial equations by interpreting the graphs of the corresponding functions, using graphing calculators or graphing software;

Planning Notes

· For Part B, students need access to a graphing calculator or graphing software, as well as graph paper.

· If students are allowed to discuss strategies at the start of Part B, the teacher should arrange groups of four students ahead of time.

Teaching/Learning Strategies

Teacher Facilitation

Part A could consist of a pencil-and-paper test containing questions similar to those posed in the activities. Emphasis should be on expectations NOT specifically covered in Part B and should be assessed primarily for Knowledge/Understanding, and Application. Sample questions are provided in the Student Activity, although equations of functions and marking scheme are left to the discretion of the individual teacher. Notes to the teacher are shown in parentheses.

In Part B, the teacher arranges students in pre-assigned groups and introduces the activity by reviewing the different forms of polynomial functions – as a table (numerical model), an equation (algebraic model), or graph (graphical model). Instructions should be reviewed carefully with students so they understand what is expected. Allow students to discuss strategies for 5-10 minutes (without allowing any writing of solutions to occur), then, students write up the activity individually. You might allow students to use their books. There are multiple ways of (correctly) justifying matches or of obtaining graphs or equations. (Note: finding the table of values given the equation is meant to be as straightforward as it seems.) The teacher should remind students that there are many ways to get a partial or approximate graph/equation/table and they should be promoted (as a good start), but it should also be made clear that accuracy of answers will be part of the assessment criteria.

Teacher Notes

In Part B, there are six of each of the types of models shown (algebraic, graphical, numerical). For each set of six, two match up with each of the other types of models, leaving two of each type of model for which the remaining forms must be generated by the student). Answers are provided at the end of the activity. To shorten this part of the activity and reduce the difficulty, the teacher could remove some or all of the extra models and/or require students only to determine the missing model for the matching pairs.

Student Activity

Part A – This is Just a Test (Sample Questions)

1. Classify each of the following functions as even, odd, or neither. [Given equations, graphs, and tables.]

2. What type of symmetry, if any, would exist for the graphs of the following functions?

3. What is the degree of the polynomial represented by the following sets of data?

4. Find the y-intercept for the polynomials in question #3. [If covered as extension.]

5. Determine the leading coefficient for the polynomials in question #3. [If covered as extension.]

6. The zeros of a function are given. Determine the general equation of the function.

7. The zeros of a function and a point on the function are given. Determine the equation of the function.

8. Determine the equations of the functions corresponding to the graphs shown.

9. For each of the following functions, complete the statements “as ____ ”
and “as ____ ” [or state whether the function starts high/low and ends high/low – provide some equations, some graphs]

10. The graphs of a number of functions and their corresponding equations are shown on the axes below. Label each graph with the appropriate equation.

11. State the zeros of the following functions: [Given equations, graphs, and tables.]

12.