chapter 5: polynomial functions - mr....

Chapter 5: Polynomial Functions Section 5.1

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Chapter 5: Polynomial Functions

Section 5.1: Exploring the Graphs of Polynomial Functions

Terminology:

Polynomial Function:

A function that contains only the operations of multiplication and addition with

real-number coefficients, whole-number exponents, and two variables. Ex: 𝑓(𝑥) = 5𝑥3 − 3𝑥 + 7

Types of Functions

1. Constant Function: A function of zero degree, whose greatest exponent is zero.

Ex: 𝑓(𝑥) = 7

2. Linear Function: A polynomial function of the first degree, whose greatest exponent is one.

Ex: 𝑓(𝑥) = 3𝑥 + 2

3. Quadratic Function:

A polynomial function of the second degree, whose greatest exponent is two.

Ex: 𝑓(𝑥) = 7𝑥2 + 4𝑥 − 3

4. Cubic Function:

A polynomial function of the third degree, whose greatest exponent is three.

Ex: 𝑓(𝑥) = 5𝑥3 + 𝑥2 − 4𝑥 + 1


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Characteristics of a Polynomial Functions

X-Intercepts:

The point(s) at which a function crosses the x-axis.

Y-Intercepts:

The point(s) at which a function crosses the y-axis.

Domain:

All the possible values of x which corresponds to a function.

Range:

All the possible values of y which corresponds to a function.

End Behaviour:

The description of the shape of the graph, from left to right

on the coordinate plane.

NOTE: The Cartesian Plane is broken into four quadrants

Ex. Given this graph,

the end behaviour is from

Quadrant III to Quadrant I

Turning Point:

Any point where the graph of a function changes form increasing to decreasing of

vice versa.

This graph has two turning points since the

y-values change from increasing to decreasing

to increasing again.

x

y

Quadrant IQuadrant II

Quadrant III Quadrant IV

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


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For each polynomial function, identify:

1. x-intercepts

2. y-intercepts

3. end behaviour

4. domain

5. range

6. number of turning points

(a)

(b)

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


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(c)

(d)

(e)

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


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Summary of Functions

Practice Questions:

1-3 pg 277


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Section 5.2: Exploring the Graphs of Polynomial Functions

Terminology:

Standard Form:

The standard form for a linear function is: 𝑓(𝑥) = 𝑎𝑥 + 𝑏, 𝑥 ≠ 0

The standard form for a quadratic function is: 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, 𝑥 ≠ 0

The standard form for a cubic function is: 𝑓(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑, 𝑥 ≠ 0

Leading Coefficient:

The coefficient of the term with the greatest degree in a polynomial function in

standard form.

Ex. For the function 𝑓(𝑥) = 2𝑥3 + 7𝑥, the leading coefficient is 2.

Constant Term:

The term with no variable in a polynomial function.

Ex. For the function 𝑓(𝑥) = 2𝑥3 + 7𝑥 + 5, the constant term is 5.

Reasoning the Characteristics of a Graph of a Given

Polynomial Function Using its Equation

Determine the following characteristics of each function using its equation.

- Number of possible x-intercepts

- y-intercepts

- end behaviour

- domain

- range

- number of possible turning points

(a) 𝑓(𝑥) = 3𝑥 − 5 (b)𝑓(𝑥) = −2𝑥2 − 4𝑥 + 8


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(c) 𝑓(𝑥) = 2𝑥3 + 10𝑥2 − 2𝑥 − 10 (d) 𝑓(𝑥) = 3𝑥2 + 5𝑥 + 4

(e) 𝑓(𝑥) = −2𝑥 + 7 (f) 𝑓(𝑥) = −5𝑥3 + 2𝑥 − 1

NOTES:


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Connecting Polynomial Functions to their Graphs

Match each graph with the correct polynomial function. Justify your reasoning.

𝑔(𝑥) = −𝑥3 + 4𝑥2 − 2𝑥 − 2 𝑗(𝑥) = 𝑥2 − 2𝑥 − 2

𝑝(𝑥) = 𝑥3 − 2𝑥2 − 𝑥 − 2 ℎ(𝑥) = −1

2𝑥 − 3

𝑘(𝑥) = 𝑥2 − 2𝑥 + 1 𝑞(𝑥) = −2𝑥 − 3

(i) (ii)

(iii) (iv)

(v) (vi)

x- 6 - 4 - 2 2 4 6

y

- 6

- 4

- 2

2

4

6

x- 8 - 6 - 4 - 2 2

y

- 8

- 6

- 4

- 2

2

x- 4 - 2 2 4 6

y

- 2

2

4

6

8

x- 4 - 2 2 4 6

y

- 4

- 2

2

4

6

x- 4 - 2 2 4 6

y

- 6

- 4

- 2

2

4

x- 4 - 2 2 4

y

- 4

- 2

2

4


125


126

Sketching a Graph Given Some Polynomial Functions

Sketch the graph of a possible polynomial function for each set of characteristics below.

What can you conclude about the equation of the function with these characteristics?

WRITE A POSSIBLE EQUATION FOR EACH!!!

(a) Range: {𝑦|𝑦 ≥ −2, 𝑦 ∈ 𝑅}

y-intercept: 4

(b) Range: {𝑦|𝑦 ∈ 𝑅}

Turning Points: One in Quadrant III and another in Quadrant I

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


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(c) End Behaviour: Quadrant III to Quadrant IV

X-intercept at (-3,0) and (1,0)

(d) Two Turning Points

End Behaviour: Quadrant II to Quadrant IV

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


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Single, Double, and Triple Roots

There are three different types of roots:

When a graph passes directly through the x-axis, we call that a single root.

When a graph touches the x-axis and then bounces back, never passing through, we call

that a double root.

When a graph levels off when approaching the x-axis, passes through and then begins

curving again, we call this a triple root (triple roots are only possible in cubic functions

and those with higher degrees)

Practice Questions:

1, 2,3 pg 287

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


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Section 5.3: Modelling Data with a Line of Best Fit

Terminology:

Scatter Plot

A set of points on a grid, used to visualize a relationship or possible trend in the

data.

Ex:

Line of Best Fit:

A straight line that best approximates the trend in a scatter plot.

Regression Function:

A line or curve of best fit, developed through a statistical analysis of data.

Interpolate:

The process used to estimate a value within the domain of a set of data based on a

trend.

Extrapolate:

The process used to estimate a value outside the domain of a set of data based on

a trend. Requires the extending of the line of best fit.

x2 4 6 8 10

y

2

4

6

8

10


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Determining a Linear Model for Continuous Data

Ex. The one hour record is the farthest distance travelled by bicycle in 1 h. The table

below shows the world record distances and the dates they were accomplished.

Year 1996 1998 1999 2002 2003 2004 2007 2008 2009 Distance

(km) 78.04 79.14 81.16 82.60 83.72 84.22 86.77 87.12 90.60

(a) Use technology to create a scatter plot and to determine the equation of the

line of best fit.

(b) Interpolate a possible world record distance for the year 2006, to the nearest

hundredth of a kilometre.

(c) Compare your estimate with the actual world record distance of 85.99 in

2006.

(d) Interpolate a possible world record distance in the year 2000, what would you

expect this distance to have been?

x5 10 15 20

y

10

20

30

40

50

60

70

80

90

100


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Using Linear Regression to Solve a Problem that Involves Discrete Data

Matt buys T-shirts for a company that prints art on T-shirts and then resells them.

When buying the T-shirts, the price Matt must pay is related to the size of the order.

Five of Matt’s past orders are listed in the table below.

Number of Shirts

Cost Per Shirt ($)

500 3.25 700 1.95 200 5.20 460 3.51 740 1.69

Matt has misplaced the information from his supplier about price discount on bulk

orders. He would like to get the price per shirt below $1.50 on his next order.

(a) Use technology to create a scatter plot and determine an equation for the linear

regression function that models the data.

(b) What does the slope and y-intercept of the equation of the linear regression

function represent in this context?

(c) Use the linear regression function to extrapolate the price per shirt of a 900 shirt

order.


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Using a Ruler to Estimate the Equation that Best Fit Equation

Using a ruler, determine the slope and y-intercept of the line of best fit for each data.

Use this information to create an equation for the data.

(a) (b)

Practice Questions:

1,2,3,4,5,6,7 pg 301-303


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Section 5.4: Modelling Data with a Curve of Best Fit

Terminology:

Curve of Best Fit

A curve that best approximates the trend of a scatter plot.

Using Technology to Solve a Quadratic Problem

Audrey is interested in how speed plays a role in car

accidents. She knows that there is a relationship between

the speed of a car and the distance needed to stop. She

has found the following experimental data on a reputable

website, and she would like to write a summary from the

graduation class website.

(a) Using Technology determine if this data is best

modelled by a quadratic or cubic function. How do

you know?

(b) What is the equation of the curve of best fit?

(c) Use your equation to compare the stopping distance at 30 km/h, to the nearest

tenth of a metre.

(d) Determine the stopping distance associated with the speed of 90 km/h.

Speed (km/h)

Distance (m)

90 94.4 36 17 65 49.2 56 50.3 65 43.1 24 10.9 35 14.2 55 57.3 81 76.5 83 100.3 25 9.1 25 10 77 77.8 32 14.9 76 67.3 38 21 92 111 22 5.6 31 16.8 50 40


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Using Observation Skills to Answer Questions about Data

Ex1. From the data, determine the best type of regression that should be used.

x5 10 15 20

y

10

20

30

40

50

60

70

80

90

100

x5 10 15 20

y

10

20

30

40

50

60

70

80

90

100

x5 10 15 20

y

10

20

30

40

50

60

70

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100


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2. Given the regression for a set of data is given as:

This is a cubic function with a regression equation of: 𝑦 = 0.091𝑥3 + 0.0001𝑥2 − 6.19𝑥 + 25

(a) Determine the cost associated with 10 hours of service.

(b) Determine the cost associated with 15 hours of service.

(c) Using the graph, determine the hours of service that would produce a cost of $50

(d) Using the graph, determine the hours of service that would produce a cost of $90

(e) What is the constant term from the equation? How does this relate to the graph?

Time (h)5 10 15 20

Cost ($)

10

20

30

40

50

60

70

80

90

100


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(f) When Jesse was trying to fit the data to the graph, he ran three different regressions:

Linear Regression: 𝑟2 = 0.9487215

Quadratic Regression: 𝑟2 = 0.987521

Cubic Regression: 𝑟2 = 0.987520

Do you think his decision to use a cubic equation was the best decision for this data. Why or Why not.

Explain.

chapter 5: polynomial functions - mr....

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