Section 8: Summary of Functions Section 8 β Topic 1
Comparing Linear, Quadratic, and Exponential Functions β Part 1
Complete the table below to describe the characteristics of linear functions.
Linear Functions Equation π = ππ + π Shape linear Rate of Change constant Number of π₯-intercepts π or π or infinitely many Number of π¦-intercepts π Number of vertices π Domain π π β β
Range π π β β or π π = ππππ ππππππππ
Sketch the graphs of three linear functions that show all the possible combinations above.
Complete the table below to describe the characteristics of quadratic functions.
Quadratic Functions
Equation π = πππ + ππ + π, where π β π Shape parabola
Rate of Change Not constant
Number of π₯-intercepts π, π, or π
Number of π¦-intercepts π
Number of vertices π
Domain π π = β
Range π π β₯ ππππ ππππππππ or π π β€ ππππ ππππππππ
Sketch the graphs of three quadratic functions that show all the possible combinations above.
Complete the table below to describe the characteristics of exponential functions.
Exponential Functions
Equation π = π β ππ where π β π and π > π.
Shape exponential
Rate of Change Not constant
Number of π₯-intercepts π or π
Number of π¦-intercepts π
Number of vertices π
Domain π π = β
Range π π > ππππ ππππππππ or π π < ππππ ππππππππ
Sketch the graphs of two exponential functions that show all the possible combinations above.
Consider the following tables that represent a linear and a quadratic function and find the differences.
Linear Function π₯ π(π₯)
0 5
1 7
2 9
3 11
4 13
How can you distinguish a linear function from a quadratic function? The first differences in a linear function are constant. The second differences, but not the first of a quadratic function are constant.
Quadratic Function π₯ π(π₯)
0 3
1 4
2 7
3 12
4 19
Linear Function
! " !
0 51 72 93 114 13
Quadratic Function
! " !
0 31 42 73 124 19
π
π
π
π
π
π
π
π
π
π
π
Consider the following table that represents an exponential function.
How can you determine if a function is exponential by looking at a table? There is a common ratio.
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Exponential Function
π₯ π(π₯)
0 1
1 3
2 9
3 27
4 81
5 243
Exponential Function
! " !
0 11 32 93 274 815 243
Γπ
Γπ
Γπ
Γπ
Γπ
Section 8 β Topic 2 Comparing Linear, Quadratic, and Exponential
Functions β Part 2
Letβs Practice! 1. Identify whether the following key features indicate a
model could be linear, quadratic, or exponential.
Key Feature
Linea
r
Qua
drat
ic
Expo
nent
ial
Rate of change is constant. l β β
2nd differences, but not 1st, are constant. β l β Graph has a vertex. β l β
Graph has no π₯-intercept. l l l
Graph has two π₯-intercepts. β l β
Graph has one π¦-intercept. l l l Domain is all real numbers. l l l
Range is π¦ π¦ > 0 . β β l Range is π¦ π¦ β€ 0 . β l β Range is all real numbers. l β β
Try It! 2. Determine whether each table represents a linear,
quadratic, or exponential function.
π π π π π π 0 1 0 7 0 2 1 2 3 13 1 6 2 5 6 19 2 18 3 10 9 25 3 54 4 17 15 37 4 162
o Linear o Quadratic o Exponential
o Linear o Quadratic o Exponential
o Linear o Quadratic o Exponential
BEAT THE TEST!
1. Identify whether the following real-world examples should be modeled by a linear, quadratic, or exponential function.
Real-World Example
Linea
r
Qua
drat
ic
Expo
nent
ial
Growing a culture of bacteria β β l The distance a Boeing 737 MAX can travel at a certain speed over a given period of time
l β β
Kicking a ball into the air β l β Running a race at a constant speed l β β A pumpkin decaying β β l Jumping from a high dive β l β
2. Complete the following table so that π(π₯) represents a linear function and π(π₯) represents an exponential function.
Answers Vary. Sample answers:
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π π(π) π(π)
β5 βπ π
β4 π π
β3 ππ π
β2 ππ ππ
β1 ππ ππ
Section 8 β Topic 3 Comparing Arithmetic and Geometric Sequences
The founder of a popular social media website is trying to inspire gifted algebra students to study computer programming. He is offering two different incentive programs for students. Option 1: Students will earn one penny for completing their first
math, science, or computer-related college course. The amount earned will double for each additional course they complete.
Option 2: Students will earn one penny for completing their first
math, science, or computer-related college course. For each subsequent course they complete, they will earn $100.00 more than they did for the previous course.
Write an explicit formula for each option. Option 1: ππ = π. ππ β ππUπ Option 2: ππ = π. ππ + πππ(π β π)
Compare the two scholarship options in the tables below.
Option 1 Option 2 Course Amount Course Amount
1 $0.01 1 $0.01 2 $0.02 2 $100.01 3 $0.04 3 $200.01 4 $0.08 4 $300.01 5 $0.16 5 $400.01 6 $0.32 6 $500.01 7 $0.64 7 $600.01 8 $1.28 8 $700.01 9 $2.56 9 $800.01 10 $5.12 10 $900.01 11 $10.24 11 $1,000.01 12 $20.48 12 $1,100.01 13 $40.96 13 $1,200.01 14 $81.92 14 $1,300.01 15 $163.84 15 $1,400.01 16 $327.68 16 $1,500.01 17 $655.36 17 $1,600.01 18 $1,310.72 18 $1,700.01 19 $2,621.44 19 $1,800.01 20 $5,242.88 20 $1,900.01 21 $10,485.76 21 $2,000.01 22 $20,971.52 22 $2,100.01 23 $41,943.04 23 $2,200.01 24 $83,886.08 24 $2,300.01 25 $167,772.16 25 $2,400.01
Compare the two scholarship options in the graphs below.
Option 1 is a geometric sequence.
Γ Each term is the product of the previous term and two.
Γ This geometric sequence follows a(n)
___________________ pattern.
Γ Evaluate the domain of this function. The set of Natural numbers: {π|π = π, π, π, πβ¦ }
Option 2 is an arithmetic sequence.
Γ Each term is the sum of the previous term and 100. Γ Arithmetic sequences follow a(n)
_____________________ pattern.
Γ Evaluate the domain of this function. The set of Natural numbers: {π|π = π, π, π, πβ¦ }
What can be said about the domain of arithmetic and geometric sequences? The domain is a subset of the integers.
Exponential f
Linear f
Letβs Practice! 1. Consider the two scholarship options for studying
computer science.
a. Which scholarship option is better if your college degree requires 10 math, engineering, or programming courses? Option 2
b. What if your degree requires 25 math, engineering, or programming courses? Option 1
c. Do you think that these graphs represent discrete or continuous functions? Justify your answer. Discrete, because you cannot take half of a course.
d. Do you think Option 1 would ever be offered as a scholarship? Why or why not?
Answers vary: Probably not unless there was a cap on the number of courses you could take.
Try It! 2. Pablo and Lily are saving money for their senior trip next
month. Pabloβs goal is to save one penny on the first day of the month and to triple the amount he saves each day for one month. Lilyβs goal is to save $10.00 on the first day of the month and increase the amount she saves by $5.00 each day.
a. Pabloβs savings plan is an example of a(n)
b. Lilyβs savings plan is an example of a(n)
c. Which person do you think will be able to meet his/her goal? Explain. Answers vary. Sample answer: Lily, Pabloβs amount to save would get exponentially large.
3. Circle the best answers to complete the following
statement. Arithmetic sequences follow a(n) linear | exponential |
quadratic pattern, whereas geometric sequences follow
a(n) linear | exponential | quadratic pattern, and the
domain of both sequences is a subset of the
integers | radicals | exponents.
o arithmetic sequence.
β’geometric sequence.
l arithmetic sequence. o geometric sequence. o
BEAT THE TEST! 1. On Sunday, Chris and Caroline will begin their final
preparations for a piano recital the following Saturday. Caroline plans to practice 30 minutes on the Sunday prior to the recital and increase her practice time by 30 minutes every day leading up to the recital. Chris plans to practice half of Carolineβs time on Sunday, but will double his practice time every day leading up to the recital.
Part A: List Carolineβs and Chrisβs practice times on the
tables below.
Carolineβs Practice Time
Chrisβs Practice Time
Sunday ππ Sunday ππ
Monday ππ Monday ππ
Tuesday ππ Tuesday ππ
Wednesday πππ Wednesday πππ
Thursday πππ Thursday πππ
Friday πππ Friday πππ
Part B: Compare the graphs of Carolineβs and Chrisβs practice times. Identify each graph as linear or exponential.
_______________________ _______________________
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Linear f
Exponential f
Section 8 β Topic 4 Exploring Non-Arithmetic, Non-Geometric Sequences
Letβs review sequences. Sequences are _____________ whose domain is a subset of the ____________. We have studied arithmetic and geometric sequences. They can both be defined by explicit and recursive formulas. Can you think of a sequence that is neither arithmetic nor geometric? π, π, π, π, ππ Consider the Fibonacci sequence. 1, 1, 2, 3, 5, 8, 13, β¦ What pattern do you notice? Starting with π, it is the sum of the previous two numbers. Write a recursive formula for the Fibonacci sequence. ππ = ππUπ + ππUπ ππ = ππ + ππ = π + π = π
functions f integers
f
Consider the following sequence. Complete the table for the sequence.
1, 4, 9, 16, 25, β¦
What type of function does this sequence appear to be? quadratic What would the domain of this function be restricted to? natural numbers Write an explicit formula for this sequence. ππ = ππ Letβs explore how to define a quadratic function with a recursive representation.
We can write a recursive formula as π π₯ = π π₯ β 1 plus the linear rate of change.
π π₯ = π π₯ β 1 +ππ₯ + π Use the table above to find the second difference. This will give us a linear rate of change. The second difference is π and it is a constant change.
π π(π) π π π π π π π ππ π ππ
Using the second difference as the linear rate of change, write the recursive formula.
π π₯ = π π₯ β 1 + ____ π₯ + π Now use values from the table to determine π. π = βπ Write the recursive formula for the sequence. π π = π π β π + ππ β π Use the recursive formula to find the next two terms of the sequence. π π = ππ π π = ππ Letβs Practice! 1. Consider the following sequence.
6, 9, 14, 21, 30, β¦.
a. What type of function does this sequence appear to
be? What are the domain restrictions?
Quadratic, natural numbers
b. Write the recursive formula for the function.
π π = π π β π +ππ + π π π = π π β π + ππ β π
π f
Try It! 2. On an algebra test, Maxβs answer was marked incorrect
when he was asked to define the recursive formula for the sequence β3, 3, 13, 27, 45, β¦. Maxβs work is shown below. Identify the mistake and correct Maxβs work.
π π = π π β π + ππ β π
BEAT THE TEST! 1. A recursive formula for a sequence is
π π₯ = π π₯ β 1 + 6π₯ β 3, where π 1 = 3, which of the following could also represent the sequence?
A π(π₯) = `
aπ₯a, where π₯ β π‘βπ πππ‘πππππ
B π(π₯) = 3π₯a, where π₯ β π‘βπ πππ‘πππππ C π(π₯) = `
aπ₯a, where π₯ β π‘βπ πππ‘π’πππ ππ’πππππ
D π(π) = πππ, where π β πππ πππππππ πππππππ
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Section 8 β Topic 5 Modeling with Functions
Letβs discuss the modeling cycle process. Consider and complete the following diagram that displays the modeling cycle process.
Letβs Practice! 1. The table below represents the population estimates (in
thousands) of the Cape Coral-Fort Myers metro area in years since 2010. Employ the modeling cycle to create a graph and a function to model the population growth. Use the function to predict the population in 2020.
a. What are the variables in this situation and what do
they represent?
π = years since 2010 π π = population estimates in thousands
Problem Report
π 0 1 2 3 4 5 6
π(π) 619 631 645 661 679 699 721
Problem β Identify the variables in the situation and select those that represent essential features.
Formulate f
Validate f
Compute f
Interpret
b. Determine what type of function models the context.
Quadratic function
c. Sketch the graph and find the function that models
the table.
Population of Cape Coral-Fort Myers Metro Area
π π = ππ + πππ + πππ
Popu
latio
n Es
tima
tes
(in T
hous
ands
)
Years since 2010
Formulate a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables.
Compute β Analyze and perform operations on these relationships to draw conclusions.
d. Use the model to predict the population in the year 2020. πππ, πππ
e. What do the results tell you about the population
growth in Cape Coral-Fort Myers metro area as it relates to the original table?
Growing at a relatively fast rate
f. What methods can we use to validate the
conclusions? π π = πππ
g. What key elements should be included in your report?
Function that we found to model, population estimate for 2020, validation and interpretation
Validate the conclusions by comparing them with the situation, and then either improve the model, or, if it is acceptable, move to the reporting phase.
Report on the conclusions and the reasoning behind them.
Interpret the results of the mathematics in terms of the original situation.
Try It! 2. According to Floridaβs Child Labor Law, minors who are
14 or 15 years old may work a maximum of 15 hours per week, and minors that are 16 or 17 years old may work a maximum of 30 hours per week. The relationship between the number of hours that a 15-year old minor in Florida works and his total pay is modeled by the graph below. What is the maximum amount that he can earn in a week?
Phase 1: _____________
a. Identify the variables in the situation and what they represent. π = hours worked π π = amount earned in dollars
Phase 2: _____________
b. What type of function can be represented by this graph? Linear function
c. Describe the end behavior of the graph. As π increases, π π increases
Hours Worked
Am
ount
of E
arni
ngs
Total Pay
Problem
Formulate
d. What does the end behavior tell you about the function? The more hours he works the more earnings he make
Phase 3: _____________
e. What strategy will you use to create the model for this situation? π π = ππ + π, find π and π
f. Find the function of the graph.
π π = ππ Phase 4: _____________
g. Complete the following statement.
The domain that best describes this situation is
{π₯|π₯ π
h. What constraints on the domain would exist for a 14-year old? A 17-year old? 14: π π β€ π β€ ππ 17: π π β€ π β€ ππ
i. How much does the student make per hour? Justify
your answer algebraically. $π, this is the slope of the function
l rational numbers}. o natural numbers}. o whole numbers}.
Compute
Interpret
Phase 5: _____________
j. Verify that your function accurately models the graph.
π ππ = πππ
k. Are there other ways to validate your function?
Choose other points
Phase 6: _____________
l. What would you report?
The equation that models the function is linear. π π = ππ was determined to be the relationship where π = hours worked and π π = amount earned in dollars. For a ππ-year-old, they can work a maximum of ππ hours per week. Thus, π ππ = $πππ is the maximum amount to earn in a week.
Validate
Report
BEAT THE TEST!
1. Dariel employed the modeling cycle to solve the following problem. Hannahβs uncle works at the BMW plant in Spartanburg, South Carolina. He purchased a 2017 BMW M2 for Hannah at the manufacturerβs suggested retail price (MSRP) of $52,500. Suppose over the next ten years, the car will depreciate an average of 9% per year. Hannah wishes to sell the car when it is valued at $22,000. When should she sell the car? When Dariel got to the compute phase, he knew something was wrong. His work is shown below.
Part A: Critique his reasoning and give feedback on where he went wrong.
The function should have been π π = πππππ(π. ππ)π so that between year 9 and 10 the value is $ππ, πππ and $ππ, πππ, respectively.
Part B: Complete the modeling cycle.
The function that models the value of the car is π π = πππππ(π. ππ)π, where π is the number of years Hannah has owned the car. If she wants to sell it when it is still worth $ππ, πππ, she should sell shortly after 9 years of ownership.
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Section 8 β Topic 6 Understanding Piecewise-Defined Functions
What is a piecewise function?
Γ A function made up of distinct β________________β
based on different rules for the _________________. Γ The βpiecesβ of a piecewise function are graphed
together on the same coordinate plane. Γ The domain is the ______________________, or the
π₯-values. Γ The range is the ____-values, or output. Γ Since it is a function, all βpiecesβ pass the vertical line
test.
Describe an example of a piecewise function used in our daily lives.
Traveling at different speeds over intervals of time on a road trip. Getting paid hourly pay at minimum wage and then getting paid overtime rate.
pieces
domain
input
π
Consider the following piecewise-defined function.
π π₯ = π₯a β 2,when π₯ β€ 02π₯ + 1,when π₯ > 0
Γ Each function has a defined ___________ value, or rule.
o π₯ is less than or equal to zero for the first function. o π₯ is greater than zero for the second function.
Γ Both of these functions will be on the same graph.
They are the βpiecesβ of this completed piecewise-defined function.
Label the βpiecesβ of π π₯ above.
domain
π(π) = ππ + π
π(π) = ππ β π
Letβs note some of the features of the graph.
Γ The domain of the piecewise graph can be represented with intervals. If we define the first interval as π₯ β€ 0, the second interval would be ____________________.
Γ The graph is nonlinear (curved) when the domain is
______________________. Γ The graph is linear when the domain is
_________________. Γ There is one closed endpoint on the graph, which
means that the particular domain value, zero, is _________________ in that piece of the function. This illustrates the inclusion of zero in the function ______________________.
Γ There is one open circle on the graph, which means
that the particular value, zero, is ________ _______________ in that piece of the function. This illustrates the constraint that π₯ > 0 for the function ________________.
π > π
π β€ π
π > π
included
ππ β π
not
included
ππ + π
Letβs Practice! 1. Airheadz, a trampoline gym, is open seven days a week
for ten hours a day. Their prices are listed below:
Two hours or less: $15.00 Between two and five hours: $25.00 Five or more hours: $30.00
The following piecewise function represents their prices:
π π₯ =15,when 0 < π₯ β€ 225,when 2 < π₯ < 5 30,when 5 β€ π₯ β€ 10
Graph the above function on the following grid.
Γ π π₯ is a special type of piecewise function known as a _____________ function, which resembles a series of steps.
Γ Step functions pair every π₯-value in a given interval
(particular section of the ________________) with a single value in the range (______-value).
Try It! 2. Consider the previous graph in exercise 1.
a. How many pieces are in the step function? Are the
pieces linear or nonlinear? Three pieces and linear
b. How many intervals make up the step function? What are the interval values? Three intervals: π, π , π, π , [π, ππ]
c. Why are open circles used in some situations and closed circles in others? In open circles number is not included in the interval and in closed circles numbers are included in interval.
d. How do you know this is a function? Vertical line test, for every input there is a unique output.
e. What is the range of this piecewise function? ππ, ππ, ππ
step
domain π
BEAT THE TEST! 1. Evaluate the piecewise-defined function for the given
values of π₯ by matching the domain values with the range values.
π π₯ =π₯ β 1, π₯ β€ β2 2π₯ β 1, β 2 < π₯ β€ 4 β3π₯ + 8, π₯ > 4
Answers: βπ,βπ , βπ,βπ , π, βπ , π, π , π, π , (π, βππ)
8β242β50
73β3β16β6β1
+ ,(+)
2. Complete the following sentences by choosing the correct answer from each box.
Part A: Piecewise-defined functions are represented by
that must
correspond to
Part B: When evaluating piecewise-defined functions, choose which equation to use based on the
then substitute and evaluate using
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o one function o at least one function l at least two functions
ldifferent domain values o different range values o real numbers
o constant l π₯-value o slope
o exponent rules l order of operations o your instincts
Section 8 β Topic 7
Absolute Value Functions Consider the absolute value function: π π₯ = π₯ . Sketch the graph of π π₯ by completing the table of values. Absolute value functions are a special case of a _______________________ function. Write a different function that represents π π₯ .
π π = βπ, ππππ π β€ ππ, ππππ π > π
π π(π)
β2 π
β1 π
0 π
1 π
2 π
piecewise
Letβs Practice! 1. Sketch the graph of π(π₯) = π₯ + 1.
Write π(π₯) as a piecewise-defined function.
π π = βπ + π, ππππ π β€ π
π + π, ππππ π > π
2. Consider the function below.
π‘ π₯ = π₯ + 2,π€βππ π₯ < β2βπ₯ β 2,π€βππ π₯ β₯ β2
Write the absolute value function that represents π‘(π₯). π π = β π + π
π π(π)
β2 π
β1 π
0 π
1 π
2 π
Try It! 3. Sketch the graph π π₯ = π₯ β 2 + 3.
Write π(π₯) as a piecewise-defined function.
π π = βπ + π, ππππ π < ππ + π, ππππ π β₯ π
4. Compare and contrast β(π₯) and π(π₯).
β π₯ = π₯ β 1 π π₯ = π₯ β 1,π€βππ π₯ < 0
βπ₯ β 1,π€βππ π₯ β₯ 0
Both are absolute value and piecewise functions Both are transformations from π π = π :
π(π) is shifted one unit to the right π(π) is shifted down one unit and reflected.
BEAT THE TEST!
1. Consider the following piecewise-defined function.
π π₯ = βπ₯ β 3,when π₯ < 0π₯ β 3,when π₯ β₯ 0
Which of the following functions also represent π(π₯)? A π π₯ = π₯ β 3 B β π₯ = π₯ + 3 C π π = π β π D π π₯ = π₯ + 3
Answer is C
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Section 8 β Topic 8 Graphing Power Functions β Part 1
A power function is a function in the form of π π₯ = ππ₯{, where π and π represent the set of all real numbers. A square root function is an example of a ______________ function. Sketch the graph of π π₯ = π₯ on the set of axes below.
Describe the domain and range.
Domain: π π β₯ π Range: π π β₯ π
power
A ____________ ____________ is a number that multiplies by itself three times in order to create a cubic value. A function is called a cube root function if _________________. Sketch the graph of π(π₯) = π₯| on the set of axes below.
Describe the domain and range. Domain: π π β β or (ββ,β) Range: π π β β or (ββ,β)
cube function
π(π) = βππ
Letβs Practice! 1. Consider the following function.
π π₯ = π₯ + 2
a. Sketch the graph of π π₯ on the set of axes below.
b. Describe the domain and range.
Domain: π π β₯ π or [π,β) Range: π π β₯ π or [π,β)
c.
2. Consider the following function.
π π₯ = π₯| β 2
a. Sketch the graph of π π₯ on the set of axes below.
b. Describe the domain and range.
Domain: π π β β or (ββ,β) Range: π π β β or (ββ,β)
Try It! 3. Consider the following functions.
β π₯ = π₯ β 1 π π₯ = π₯| β 1
a. Sketch the graphs of β π₯ and π π₯ on the set of axes
below.
b. Compare and contrast the graphs of β(π₯) and π(π₯).
Both are power functions π(π) is a square root function π(π) is a square root function They have different domains and ranges
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Section 8 β Topic 9 Graphing Power Functions β Part 2
A cubic function is any function of the form π π = ππ. Sketch the graph of π π₯ = π₯` on the set of axes below.
Describe the domain and range. Domain (ββ,β) Range (ββ,β)
Letβs Practice! 1. Consider the following function.
π π₯ = π₯` + 1
a. Sketch the graph of π π₯ on the set of axes below.
b. Describe the domain and range. Domain (ββ,β) Range (ββ,β)
c. Try It!
2. Consider the following functions.
π π₯ = π₯` β π₯ = π₯|
a. Sketch the graphs of π π₯ and β π₯ on the set of axes
below.
b. What observations can you make about the relationship between π(π₯) and β(π₯)?
π(π) increases and decreases at a much faster rate.
BEAT THE TEST!
1. Consider the following functions.
π π₯ = π₯ π π₯ = π₯| π‘ π₯ = π₯`
Which of the following statements is correct about the graph of π(π₯), π(π₯), and π‘(π₯)? C
A The domain of π(π₯), π(π₯), and π‘(π₯) is all real numbers. B The range of π(π₯) and π‘(π₯) is [3,β). C The range of π(π) is [π,β). D π(π₯), π(π₯), and π‘(π₯) share the points 0, 0 , (1, 1), and
(β1,β1).
2. A cubeβs volume, π(π ), is given by the equation π(π ) = π `, where π is the length of each side.
Part A: Sketch the graph that models the relationship
between the volume and length of each side of a cube.
Part B: What are the domain and range of this relation?
Domain (π,β) Range (π,β)
Part C: If we know the volume of a cube, write a function π (π) that we can use to find the length of the sides of the cube.
π½(π) = ππ π(π½) = π½π
Part D: How would the graph of π (π) compare to the graph of π(π )?
π½(π) is a cubic function. π(π½) is a cube root function.
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Section 8 β Topic 10 Finding Zeros of Polynomial Functions of
Higher Degrees How can you find zeros when given the graph of a polynomial function? Find the π-intercepts. How can you find zeros when given the equation of a polynomial function in factored form? Use the zero product property and set each factor equal to zero and solve. How do you determine if π₯ is a solution or zero for π π₯ ? If π is a solution, then π π = π. Consider the following graph of π(π₯).
What are the zeros of π(π₯)? π = βπ , π = βπ, and π = π
π¦
π₯
Consider the following fourth degree polynomial function.
π π₯ = π₯οΏ½ β 4π₯a Find the range of π(π₯) for the given domain {β2,β1, 0, 1, 2}. π βπ = βπ π β π βπ π = ππ β π π = π π βπ = βπ π β π βπ π = π β π = βπ π π = ππ β π π π = π π π = ππ β π π π = π β π = βπ π π = ππ β π π π = ππ β π π = π Range: {π, βπ} Does the above domain contain zeros of π(π₯)? Justify your answer. Yes, βπ, π, and π are all zeros because for those values π π = π. Consider the following third degree polynomial function.
β π₯ = βπ₯` β 5π₯a Find the zeros of the function β(π₯). βππ β ππ = π βππ π + π = π βππ = π or π + π = π π = π or π = βπ
Letβs Practice! 1. Consider the following graph of π(π₯).
What are the zeros of π(π₯)? π = βπ, π = π, and π = π.
2. What are the zeros of π π₯ = π₯(π₯ + 1)(π₯ β 2)a?
π = π, π + π = π, π β π π = π
π = π, π = βπ, π = π
π¦
π₯
Try It! 3. Consider the function β π₯ = π₯` β 3π₯a + 2.
a. Find the range of β(π₯) given domain {β1, 1, 3}.
π βπ = βπ π β π βπ π + π = βπ β π π + π = βπ π π = π π β π π π + π = π β π π + π = π π π = π π β π π π + π = ππ β π π + π = ππ β ππ + π = π {βπ, π, π}
b. Are any zeros of β(π₯) found in the above domain? Justify your answer. Yes, π = π is a zero because π π = π.
c. Consider the graph of β(π₯).
What are the other zeros of β(π₯)?
π = βπ. ππ, and π = π. ππ
π¦
π₯
BEAT THE TEST! 1. Which of the graphs has the same zeros as the function
π π₯ = 2π₯` + 3π₯a β 9π₯? A
B
C
D
Answer: B
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π¦
π₯
π¦
π₯
π¦
π₯
π¦
π₯
l
Section 8 β Topic 11 End Behavior of Graphs of Polynomials
Make observations about the end behavior of the following graphs.
π¦ = π₯a As π increases, π increases. As π decreases, π increases.
π¦ = βπ₯a As π increases, π decreases. As π decreases, π decreases.
π¦
π₯
π¦
π₯
π¦ = π₯`
As π increases, π increases. As π decreases, π decreases.
π¦ = βπ₯` As π increases, π decreases. As π decreases, π increases.
π¦
π₯
π¦
π₯
Use your observations to sketch the graphs and make conjectures to complete the table.
End Behavior of Polynomials
Leading Coefficient is Positive
Leading Coefficient is Negative
Degree of
Polynomial is Even
π π₯ = π₯a
π π₯ = βπ₯a
As π₯ β β, π(π₯)__________ As π₯ β ββ, π(π₯)________
As π₯ β β, π(π₯)__________ As π₯ β ββ, π(π₯)________
Degree of
Polynomial is Odd
π π₯ = π₯`
π π₯ = βπ₯`
As π₯ β β, π(π₯)__________ As π₯ β ββ, π(π₯)________
As π₯ β β, π(π₯)__________ As π₯ β ββ, π(π₯)________
β β
β β
β β β β
β ββ
β ββ
β ββ
β ββ
Letβs Practice! 1. Consider the following graph of π(π₯).
a. Does the function π(π₯) have an even or odd degree? Justify your answer. Even. The graph opens in the same direction on both sides.
b. Is the leading coefficient of π(π₯) positive or negative? Justify your answer.
Negative. As π β β, π π β ββ and As π β ββ, π π β ββ
2. Describe the end behavior of the function
π π₯ = β5π₯` + 8π₯a β 9π₯. As π β β, π π β ββ and As π β ββ, π π β β
Try It! 3. Consider the following graph of π(π₯).
a. Does the function π(π₯) have an even or odd degree?
Justify your answer. Odd. The graph opens in opposite directions on the two sides.
b. Is the leading coefficient of π(π₯) positive or negative? Justify your answer.
Positive. As π β β, π(π) β β and As π β ββ, π π β ββ
4. Describe the end behavior of the function below.
π π₯ =12 π₯
οΏ½ β π₯οΏ½ β π₯οΏ½ + 2π₯` β 2π₯ + 2
As π β β, π(π) β β and As π β ββ, π π β β
π¦
π₯
BEAT THE TEST! 1. Determine which of the following statements is true for the
function π(π₯) = 3π₯οΏ½ + 7π₯ β 4247. A As π₯ β β, π(π₯) β β and as π₯ β ββ, π(π₯) β β. B As π₯ β β, π π₯ β ββ and as π₯ β ββ, π π₯ β ββ. C As π₯ β β, π π₯ β ββ and as π₯ β ββ, π(π₯) β β. D As π β β, π(π) β β and as π β ββ, π π β ββ.
Answer is D.
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Section 8 β Topic 12 Graphing Polynomial Functions of Higher Degrees
Consider the following function.
π π₯ = β(π₯ + 3)(π₯ β 1)(π₯ β 2) Describe the end behavior of the graph of π π₯ . As π β β, π π β ββ and As π β ββ, π π β β Find the zeroes ofπ π₯ . π = βπ, π = π, and π = π Use the end behavior and zeroes to sketch the graph of π(π₯).
Letβs Practice! 1. Sketch the graph of the following polynomial.
π π₯ = (π₯ β 2)(π₯ + 3)(π₯ + 5)
End Behavior: As π β β, π π β β and as π β ββ, π π β ββ Zeros: π = π, π = βπ, and π = βπ
2. Sketch the graph of the following polynomial.
π π₯ = β π₯ β 5 π₯ + 4 3π₯ β 1 (π₯ + 2)
End Behavior: As π β β, π π β ββ and as π β ββ, π π β ββ Zeros: π = π, π = βπ, π = π
π, and π = βπ
Try It!
3. Sketch a graph of the following polynomial.
π π₯ = (π₯ β 1)(π₯ + 2)(π₯ β 3)(π₯ + 1)
End Behavior: As π β β, π π β β and as π β ββ, π π β β Zeros: π = π, π = βπ, π = π, and π = βπ
BEAT THE TEST!
1. Match each equation with its corresponding graph. A. π¦ = (π₯ + 1)(π₯ β 3)(π₯ + 2) B. π¦ = β(π₯ + 1)(π₯ β 3)(π₯ + 2)
C. π¦ = βπ₯(π₯ + 1)(π₯ β 3)(π₯ + 2) D. π¦ = π₯(π₯ + 1)(π₯ β 3)(π₯ + 2)
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A D
C B
Section 8 β Topic 13 Recognizing Even and Odd Functions
An even function has symmetry about the ____________. An odd function has symmetry about the ____________. Consider the following graphs. Label each graph as even, odd, or neither in the space provided.
Even Odd
Neither Odd
π-axis
origin
If a function is even, then π βπ₯ = ________. If a function is odd, then π βπ₯ = ________. To determine if a function is even or odd
Γ Substitute (βπ₯) into the function. Γ If the resulting polynomial is the same, then the
function is_______. Γ If the resulting polynomial is the exact opposite, then
the function is ________. Γ If the resulting polynomial is neither the same nor the
exact opposite, the function is not even nor odd. Letβs Practice! 1. Complete the table below to determine if the following
functions are even, odd, or neither.
Function Value of π(βπ) Even, Odd, or Neither?
π π₯ = π₯οΏ½ + π₯` π βπ = βπ π + βπ π
= ππ β ππ Neither
π π₯ = π₯οΏ½ + 1 π βπ = βπ π + π = ππ + π Even
π π₯ = 6π₯οΏ½ β π₯` π βπ = π βπ π β βπ π
= βπππ + ππ Odd
odd
even
βπ(π)
π(π)
Try It! 2. Give an example of a polynomial function that is an even
function.
π π = ππ + π π βπ = ππ + π
3. Give an example of a polynomial function that is an odd
function.
π π = ππ + π 4. Give an example of a polynomial function that is neither
odd nor even.
π π = ππ + π π βπ = βππ + π π π = ππ + ππ π βπ = ππ β ππ
BEAT THE TEST! 1. Use the following functions to complete the table below.
π π₯ = π₯a π π₯ = π₯`
β π₯ = π₯` β 4π₯
Function Even Odd Neither
π(π₯) β π(π₯) o l o π(π₯) β β(π₯) l o o π π₯ + β(π₯) o o l π π₯ + β(π₯) o l o
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Section 8 β Topic 14 Solutions to Systems of Functions
Consider the functions π(π₯) and π(π₯). What can be said about the π₯-coordinates where π π₯ intersects π(π₯)? The same π-value Name two methods of finding the solutions to the system containing π(π₯) and π(π₯). Look at graph and find intersection. Find where π π = π(π) Letβs Practice! 1. Given the exponential function, π π₯ = 2οΏ½, and the cubic
function, π π₯ = π₯` β 2π₯, find the positive π₯ value where π π₯ = π(π₯) by completing a table of values from π₯ = 0 to π₯ = 4.
π π(π) π(π)
π π π
π π βπ
π π π
π π ππ
π ππ ππ
2. Given the graphs of an absolute value function, π π₯ = 4 β π₯ , and a logarithmic function, π π₯ = πππ`π₯, find the coordinate where π π₯ = π π₯ . (π, π)
!(#)
%%#%
&
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%%#%
&
Try It! 3. It takes Samara three hours to complete a job. Samara
often works with Frederick. The function that models the time that it takes them to complete the job together is given by π π₯ = `οΏ½
οΏ½οΏ½`, where π₯ represents the number of
hours it takes Frederick to complete the job alone. The graph of π π₯ is shown below.
a. Should the domain of π(π₯) be restricted? Yes, {π|π β₯ π}
b. Sketch the graph of π₯ = 3 on the same coordinate plane and interpret the meaning of the intersection in the given context. If it takes Frederick three hours, they can complete the job together in an hour and a half.
c. Sketch the graph of π₯ = 7 on the same coordinate
plane and interpret the meaning of the intersection in the given context. If it takes Frederick seven hours, they can complete the job together in two hours, six minutes.
!(#)
%%#%
&
BEAT THE TEST! 1. Given the exponential function π π₯ = 4 β 3οΏ½ and the
polynomial function π π₯ = π₯οΏ½ + 5π₯a, find the coordinate where π π₯ = π(π₯). (π, ππ)
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