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?