Unit 1: Polynomial Functions 1.0 Prerequisite Skills

Function Notation Ex 1: Determine each value for the function, 43)( 2 xxxf . a) f(2) c) f(1/4)

b) f(-1) d) f(a+1)

Equation of a Line Ex 2: Determine the equation of a line:

a) with slope -2/3 that passes through (5,-2). b) that passes through the points (2,3) and (-1,-12). c) with x-intercept 5 and y-intercept -8.

Slope and y-intercept of a Line Ex 3: State the slope and y-intercept of each line.

a) 3y = 9 - 6x b) 2x + 8y – 3 =0

MHF 4U 1-1

Finite Differences Ex 4: Use finite differences to determine if each function is linear, quadratic, or neither.

a) x y First difference Second difference

-2 1 -1 -2 0 -3 1 -2 2 1 3 6 4 13

b) x y First difference Second difference

-3 20 -2 3 -1 -2 0 -1 1 0 2 -5 3 -22

Transformations Ex 5: State the transformations that are applied to the function y=f(x) resulting in the given

transformed function. a) y=f(x-3)+2

b) y=-2f(0.5x)

c) y=0.5f(-3x)

MHF 4U 1-2

Domain and Range Ex 6: Sketch each graph and state the domain and range. a) y=2(x+1)²-3 b) 42 xy c) 12 xy

Domain and Range a) D: b) D: c) D: R: R: R: Quadratic Functions Ex 7: Determine the equation of a quadratic function has x-intercepts of 8 and -4 and passing through the point (2,-12).

MHF 4U 1-3

Ex 8: Determine the x-intercepts, the vertex, the direction of opening, and the domain and range of each quadratic function. Then, graph the function.

a) )52)(4( xxy

b) 9)3(4

1 2 xy

c) 10133 2 xxy Homework: pp 2-3#1-12

MHF 4U 1-4

MHF 4U 1-5

Unit 1: Polynomial Functions 1.1 Power Functions

POWER FUNCTION Power Function - a function of the form naxxf )( where 0a and n is a positive integer a power function is the simplest type of polynomial function and has the form

naxxf )( where x is a variable, a is a real number, and n is a whole number. Power functions have similar characteristics depending whether their degree is even or odd. Even-degree power functions have line symmetry in the y-axis. Odd-degree power functions

have point symmetry about the origin (0,0).

POLYNOMIAL EXPRESSION Polynomial expression is one or more terms where each term is the product of a constant

and a variable raised to a non-negative integral exponent only.

POLYNOMIAL FUNCTIONS A Polynomial function is a function defined by a polynomial in one variable written in the form

011

1 ...)( axaxaxaxf nn

nn

, for example, 4;1;3 32 yxyxy

Relation is a function if for every x-value there is only one y value.

the graph of the relation represents a function if it passes the vertical line test (if a vertical line drawn anywhere along the graph intersects that graph at no more than one point).

To be a polynomial function, the following conditions must be met: 1. 0na 2. the coefficients ),,...,,( 011 aaaa nn are all real numbers. 3. the exponents area all whole numbers

the degree of a polynomial function is based on the highest exponent found in that function

Domain is the set of all first coordinates of the ordered pairs of a function.

Range is the set of all second coordinates of the ordered pairs of a function.

The leading coefficient of a polynomial function is the constant belong to the power with the highest exponent.

the end behaviour of the graph is the behaviour of the y-values as x increases (that is, as x approaches positive infinity, written as x ) and as x decreases (that is, as x approaches negative infinity, written as )x .

a graph has a line of symmetry if there is a line x=a that divides the graph into two parts such that each part is a reflection of the other in the line x=a.

a graph has a point of symmetry about a point (a,b) if each part of the graph on one side of (a,b) can be rotated 180 to coincide with part of the graph on the other side of (a,b).

MHF 4U 1-6

Ex 1: a) Use a graphic calculator to graph the following functions on graph paper. b) State the degree and name of each function.

Function Degree Name of Function 1y xy

2xy 3xy 4xy 5xy 6xy

c) Complete the table.

Key Features of the Graph nxy , n is odd nxy , n is even Domain

Range

Symmetry

End Behaviour

Ex 2: Write each function in the appropriate row of the second column of the table. Give reasons for your choices.

510721586

4343 2y 3xy 4 3 2 xxyxyxyxyxyxyxy

End Behaviour Function Reason

Extends from Quadrant 3 to 1

Extends from Quadrant 2 to 4

Extends from Quadrant 2 to 1

Extends from Quadrant 3 to 4

MHF 4U 1-7

Ex 3: Determine which functions are polynomials, Justify your answer. State the degree and

the leading coefficient of each polynomial function.

Function Yes/No – Reason? Degree Leading Coefficient

12)( xxf

73)( 2 xxxg

54)( xxh

Xy cos

Ex 4: For each function

i) state the domain and range ii) describe the end behaviour iii) identify any symmetry

a) 2x2y b) x5.0y c) 3xy ii) extends from quadrant 3 to 4 ii) extends from 3 to 1 ii) extends from 2 to 4

MHF 4U 1-8

INTERVAL NOTATION Sets of real numbers may be described in a variety of ways: As an inequality, -3 < x ≤ 4

In interval notation (-3, 4] Square brackets indicate that the end value is included in the interval, and round brackets indicate that the end value is not included.

Graphically, on a number line Ex 5: Complete the chart. Bracket Interval

Inequality Number Line In Words

The set of all real numbers x such that

x is greater than-3 and less than 5

4x1

6x2

x is greater than or equal to 1

3,

0x

x is an element of the real numbers

Homework: pp 11-14 #1-12

MHF 4U 1-9

MHF 4U 1-10

Unit 1: Polynomial Functions 1.2 Characteristics of Polynomial Functions

A: Polynomial Functions of Odd Degree 1. a) Using a graphing calculator, graph each cubic function.. Group A i) 3xy ii) 4x4xxy 23 iii) 9x3x5xy 23 Group B i) 3xy ii) 4x4xxy 23 iii) 9x3x5xy 23

b) Complete the chart for the graphs in each group.

End Behaviour # of max/ min points

# of local max/min

# of x-intercepts

Group A i. ii. iii.

Group B i. ii. iii.

c) Which group of graphs is similar to the graph of

i) ?xy

ii) ?xy

MHF 4U 1-11

B: Polynomial Functions of Even Degree 1. a) Using a graphing calculator , graph each quartic function. Group A i) 4xy ii) 8x4x6xxy 234 iii) 4x11x3x3xy 234 Group B i) 4xy ii) 10x5x5xy 34 iii) 4x11x3x3xy 234

b) Complete the chart for the graphs in each group.

End Behaviour # of max/ min points

# of local max/min

# of x-intercepts

Group A i. ii. iii.

Group B i. ii. iii.

c) Explain which group of graphs is similar to the graph of

i) .xy 2

ii) .xy 2

MHF 4U 1-12

Ex 1: Determine the key features of the graphs of each polynomial function. Use these features to match each function with its graph. State the number of x-intercepts, the number of maximum and minimum points, and the number of local maximum and minimum points for the graph of each function. How are these features related to the degree of the function? a) xxxf 23 b) 55443 234 xxxxxg c) 1518372 2345 xxxxxxh d) 1010122 246 xxxxxp 1 2 3 4

MHF 4U 1-13

Ex 2: Complete the following chart:

Finite Differences For a polynomial function of degree n, where n is a positive integer, the nth differences

are equal (or constant) have the same sign as the leading coefficient are equal to ,12...1 nna where a is the leading coefficient

Ex 3: Use finite differences to determine i) the degree of the polynomial function ii) the sign of the leading coefficient iii) the value of the leading coefficient

Homework: pp 26-29#1-6, 14-17

Equation Degree Even/Odd Degree?

Leading Coefficient

x

x

# of Turning Points

a) 12 24 xxxf

b) 523 23 xxxxg

c) 243110

21 xxxxh

d) xxxf 3

e) 46 32 xxxg

f) xxxh 35

g) 432 xxxf

h) xxxxg 232 37

i) 1323 34 xxxxh

j) xxxf 2

x y -3 140 -2 37 -1 8 0 5 1 4 2 5 3 32

End Behaviour

MHF 4U 1-14

MHF 4U 1-15

Unit 1: Polynomial Functions 1.3 Equations and Graphs of Polynomial Functions

Steps for sketching polynomial functions in factored form:

1. Find y-intercept (let x = 0)

2. Find x-intercept(s) (let y = 0) and consider turning point at the x-intercept. (Even orders will “bounce” at x-axis, while odd orders will pass through.)

3. Identify the degree and leading coefficient to determine the end behavior.

ODD DEGREE [end behaviours are opposite direction] positive leading coefficient - graph starts low and ends high negative leading coefficient - graph starts high and ends low

EVEN DEGREE [ end behaviours are the same direction]

positive leading coefficient graph starts high and ends high negative leading coefficient - graph starts low and ends low

4. Select test values which are x-values between the x-intercepts or real roots. Test values

help to determine positive and negative regions on the graph and additional points to supplement sketch.

positive region (the graph above x-axis): f(x) > 0 negative region (the graph below x-axis): f(x) < 0

Example 1: a) )3)(2)(1( xxxy b) )1()2(2)( 2 xxxg

MHF 4U 1-16

c) 23 )2()1( xxy Example 2: Sketch the graph of a polynomial function that satisfies the following conditions: degree 3, positive leading coefficient, 2 zeros, 2 turning points

MHF 4U 1-17

SYMMETRY IN POLYNOMIAL FUNCTIONS EVEN FUNCTIONS The graph of an even function is symmetric in the y-axis, meaning that its graph remains

unchanged when reflected about the y-axis. An even function f satisfies f(-x) = f(x) for all x in its domain. This means that a line

segment connecting f(x) and f(-x) is a horizontal line. Examples:

1. a) Complete the table of values and graph each function. Use a graphing calculator to verify answers.

f(x) = x2 f(x) = -3x2 + 5 f(x) = x4 - 3x2 f(x) = x2 - 2x + 1

b) Which function above is not an even function? Why?

2. a) Determine whether the function is an even function or not. Use a graphing calculator to help you with your answer.

i) 24 xxy ii) 232 xxy iii) 12 26 xxy iv) 326 xxy

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

3. Determine algebraically [use property f(-x)=f(x)] whether the function is an even function or not. a) 24 2)( xxxf b) 12)( 2 xxxf

x -2 -1 0 1 2 f(x)

x -2 -1 0 1 2 f(x)

x -2 -1 0 1 2 f(x)

x -2 -1 0 1 2 f(x)

MHF 4U 1-18

ODD FUNCTIONS The graph of an odd function is symmetric about the origin, meaning that its graph remains

unchanged after rotation of 180 about the origin. An odd function f satisfies f(-x) = -f(x) for all x in its domain. This means that a line

segment connecting f(-x) and f(x) contains the origin.

Examples: 1. a) Complete the table of values and graph each function. Use a graphing calculator to

verify answers.

f(x) = x f(x) = x3 f(x) = x3 - x f(x) = x3 - 2x + 1

b) Which function above is not an even function? Why?

2. a) Determine whether the function is an odd function or not. Use a graphing calculator to help you with your answer. i) xxy 23 ii) 23 35 xxy iii) 23 xxy iv) xxxy 235

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

3. Determine algebraically [use property f(-x) = -f(x)] whether the function is an odd function

or not. a) xxxf 42)( 3 b) 1)( 3 xxxf Homework: pp 39-41#1-14

x -2 -1 0 1 2 f(x)

x -2 -1 0 1 2 f(x)

x -2 -1 0 1 2 f(x)

x -2 -1 0 1 2 f(x)

MHF 4U 1-19

MHF 4U 1-20

Unit 1: Polynomial Functions 1.4 Transformations

TRANSFORMATIONS The parameters a, k, d, and c in polynomial functions of the form cdxkay n )]([ , where n is a non-negative integer, correspond to the following transformations: a corresponds to a vertical stretch or compression and, if a<0, a reflection in the x-axis k corresponds to a horizontal stretch or compression and, if k<0, a reflection in the y-axis c corresponds to a vertical translation up or down d corresponds to a horizontal to the left or right Ex 1: a) The graph of 2xy is transformed to obtain the graph of 2)]1(3[2 2 xy

i) State the parameters and describe the corresponding transformations. a

k c d ii) Complete the table. iii) State the domain and Range. iv) State a mapping that applies transformations to the original power function

2xy 22xy 2)3(2 xy 2)13(2 2 xy

Mapping

x Mapping

y 2,-8- 4,2 1,-1- 1,1 0,0 0.0 1,1 1,1 2,8 4,2 Domain Range

MHF 4U 1-21

Ex 1: b) The graph of 3xy is transformed to obtain the graph of .1325.0 3 xy i) State the parameters and describe the corresponding transformations.

a k c d

ii) Complete the table. iii) State the domain and Range. iv) State a mapping that applies transformations to the original power function

3xy 35.0 xy 3)2(5.0 xy 1)3(2[5.0 3 xy

Mapping

x Mapping

y 2,-8- 4,2 1,-1- 1,1 0,0 0.0 1,1 1,1 2,8 4,2 Domain Range

v) Sketch a graph of .2132 2 xy and

.1325.0 3 xy on the same graph.

Homework: pp 49-52#1-16

MHF 4U 1-22

MHF 4U 1-23

Unit 1: Polynomial Functions 1.5 Slopes of Secants and Average Rate of Change

Page 1/4

RATE OF CHANGE A rate of change is a measure of the change in one quantity (the dependent variable) with respect to a change in another quantity (the independent variable). There are two types of rates of change: 1) average and ii) instantaneous. An average rate of change is a change that takes place over an interval. Instantaneous rate of change is a change that takes place in an instant. Ex 1: How can you connect average rate of change and slope?

1. Seismic activity at a certain point on the ocean floor

creates a wave that spreads in a circular pattern over the calm surface of the ocean. The table shows the radius of the circular pattern during the first 10 s as the wave moves outward. a) Identify the independent variable and the dependent

variable. Justify your choice.

b) Determine time in change

radius in changetr

for each time

interval. i) [0,10] ii) [0,1] iii) [9,10]

c) Graph the data. What type of polynomial function does the graph represent? Explain.

Time, t (s) Radius, r (m) 0 0 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20

MHF 4U 1-24

d) A secant is a line that connects two points on a curve. Draw a secant on the graph to connect the pair of points associated with each time interval in part b). What is the slope of each secant line? State the units of the secant line.

e) What is the relationship between the values found in part b) and the graph? Explain.

2. This table shows the total area covered by the wave during the first 10 s. a) Identify the independent variable and the

dependent variable. Justify your choice.

b) Determine radius in changeArea in change

rA

for each radius

interval. i) [0,20] ii) [0,4] iii) [6,12] iv) [0,2] v) [14,16]

Radius, r (m) Area, A (m²) 0 0 2 12.57 4 50.27 6 113.10 8 201.06 10 314.16 12 452.39 14 615.75 16 804.25 18 1017.88 20 1256.64

MHF 4U 1-25

c) Interpret the values found in part b). State the units for these values.

d) Graph the data. What type of polynomial function does the graph represent? Explain.

e) On the graph, draw a secant to connect the pair of points associated with each radius interval in part b). What is the slope of each secant line?

f) What is the relationship between the values found in part b) and the secant lines? How are these related to the slope of the graph? Explain.

AVERAGE RATE OF CHANGE The average rate of change between two points corresponds to the slope of the secant between the points. For example, the average rate of change of y with respect to x between the points 222111 y,xP and y,xP is determined as follows:

12

12xxyy

x in change yin change

xy

Average rate of change

MHF 4U 1-26

Calculate and Interpret Average Rates of Change From a Graph/Table of Values Ex 1: Andrew drains the water from a hot tub. The tub holds 1600 L of water. It takes 2 h for

the water to drain completely. The volume V, in litres, of water remaining in the tub at various times t, in minutes, is shown in the table and graph.

a) Determine the average rate of change for i) 30 min and 100 minutes ii) [20. 90]

b) Determine an equation that best represents the model.

Calculate and Interpret Average Rates of Change From an Equation Ex 2: A rock is tossed upward from a cliff that is 120 m above the water. The height of the

rock above the water is modeled by ,120t10t5th 2 where h(t) is the height in metres and t is time in seconds. a) Determine the average rate of change in height during each of the following time intervals.

i) [0,1] ii) [2,3]

b) Consider the graph ,120t10t5th 2 with secant lines AB and CD. Describe the relationship between the values in part a), the secant lines, and the graph.

Homework: pp 62-63 #2, 3, 6, 7, 11

Time (min)

Volume (L)

0 1600 10 1344 20 1111 30 900 40 711 50 544 60 400 70 278 80 178 90 100 100 44 110 10 120 0

MHF 4U 1-27

MHF 4U 1-28

Unit 1: Polynomial Functions 1.6 Slopes of Tangent and Instantaneous Rate of Change

RELATIONSHIP BETWEEN THE SLOPE OF SECANTS AND THE SLOPE OF A TANGENT As a point Q becomes very close to a tangent point P, the slope of the secant line becomes closer to (approaches) the slope of the tangent line. Often an arrow is used to denote the word “approaches.” So, the above statement may be written as follows: As Q->P, the slope of secant PQ-> the slope of the tangent at P. Thus, the average rate of change between P and Q becomes closer to the value of the instantaneous rate of change at P. Ex 1: During Apollo 14 mission, Alan Sheppard hit a golf ball on the Moon. The function models

the height of the golf ball’s trajectory on the Moon, where h(t) is the height, in metres, of the ball above the surface of the Moon and t is the time in seconds. How fast was the ball traveling 6 seconds after the ball was hit? i) Using a graph:

“The height of the golf ball’s trajectory on the Moon”

MHF 4U 1-29

ii) Using a table of values:

iii) Using the equation:

28.018)( ttth Homework: pp 62-63 #2, 3, 6, 7, 11

Time (s) Height (m)

0 0 1 17.2 2 32.8 3 46.8 4 59.2 5 70 6 79.2 7 86.8 8 92.8 9 97.2

MHF 4U 1-30