chapter 5: polynomial functions - mr....
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Chapter 5: Polynomial Functions Section 5.1
117
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
Chapter 5: Polynomial Functions Section 5.1
118
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
Chapter 5: Polynomial Functions Section 5.1
119
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
Chapter 5: Polynomial Functions Section 5.1
120
(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
Chapter 5: Polynomial Functions Section 5.1
121
Summary of Functions
Practice Questions:
1-3 pg 277
Chapter 5: Polynomial Functions Section 5.2
122
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
Chapter 5: Polynomial Functions Section 5.2
123
(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:
Chapter 5: Polynomial Functions Section 5.2
124
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
Chapter 5: Polynomial Functions Section 5.2
125
Chapter 5: Polynomial Functions Section 5.2
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
Chapter 5: Polynomial Functions Section 5.2
127
(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
Chapter 5: Polynomial Functions Section 5.2
128
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
Chapter 5: Polynomial Functions Section 5.3
129
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
Chapter 5: Polynomial Functions Section 5.3
130
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
Chapter 5: Polynomial Functions Section 5.3
131
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.
Chapter 5: Polynomial Functions Section 5.3
132
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
Chapter 5: Polynomial Functions Section 5.4
133
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
Chapter 5: Polynomial Functions Section 5.4
134
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
80
90
100
Chapter 5: Polynomial Functions Section 5.4
135
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
Chapter 5: Polynomial Functions Section 5.4
136
(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.