Section 3: Introduction to Functions53
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Section 3: Introduction to Functions
Topic 1: Input and Output Values ............................................................................................................................... 55
Topic 2: Representing, Naming, and Evaluating Functions ..................................................................................... 58
Topic 3: Adding and Subtracting Functions ............................................................................................................... 60
Topic 4: Multiplying Functions ...................................................................................................................................... 62
Topic 5: Closure Property .............................................................................................................................................. 66
Topic 6: Real-World Combinations and Compositions of Functions ....................................................................... 68
Topic 7: Key Features of Graphs of Functions β Part 1 .............................................................................................. 71
Topic 8: Key Features of Graphs of Functions β Part 2 .............................................................................................. 74
Topic 9: Average Rate of Change Over an Interval ................................................................................................ 77
Topic 10: Transformations of Functions ....................................................................................................................... 80
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Section 3: Introduction to Functions54
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The following Mathematics Florida Standards will be covered in this section: A-APR.1.1 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-SSE.1.1a - Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.1.2 - Use the structure of an expression to identify ways to rewrite it. For example, see π₯π₯"βπ¦π¦" as (π₯π₯')'β(π¦π¦')' thus recognizing it as a difference of squares that can be factored as (π₯π₯'βπ¦π¦')(π₯π₯' + π¦π¦'). F-BF.1.1bc - Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. Compose functions. For example, if ππ(π¦π¦) is the temperature in the atmosphere as a function of height, and β(π‘π‘) is the height of a weather balloon as a function of time, then ππ(β(π‘π‘)) is the temperature at the location of the weather balloon as a function of time.
F-BF.2.3 - Identify the effect on the graph of replacing ππ(π₯π₯) by ππ(π₯π₯) + ππ, ππππ(π₯π₯), ππ(πππ₯π₯), and ππ(π₯π₯ + ππ) for specific values of ππ (both positive and negative); find the value of ππgiven the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-IF.1.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ππ is a function and π₯π₯ is an element of its domain, then ππ(π₯π₯) denotes the output of ππ corresponding to the input π₯π₯. The graph of ππ is the graph of the equation π¦π¦ = ππ(π₯π₯). F-IF.1.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.2.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.2.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function β(ππ) gives the number of person-hours it takes to assemble ππ engines in a factory, then the positive integers would be an appropriate domain for the function. F-IF.2.6 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
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Section 3: Introduction to Functions55
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Section 3: Introduction to Functions Section 3 β Topic 1
Input and Output Values A function is a relationship between input and output.
Γ Domain is the set of values of π₯π₯ used for the ___________ of the function.
Γ Range is the set of values of π¦π¦ calculated from the
domain for the __________ of the function. In a function, every π₯π₯ corresponds to only one π¦π¦.
Γ π¦π¦ can also be written as ππ π₯π₯ . Consider the following function.
For every π₯π₯ there is a unique π¦π¦ .
input output domain range
1
2
3
25
50
75
We also refer to the variables as independent and dependent. The dependent variable ______________ ____ the independent variable. Refer to the mapping diagram on the previous page. Which variable is independent? Which variable is dependent? Consider a square whose perimeter depends on the length of its sides. What is the independent variable? What is the dependent variable? How can you represent this situation using function notation?
We can choose any letter to represent a function, such as ππ(π₯π₯) or ππ(π₯π₯), where π₯π₯ is the input value. By using different letters, we show that we are talking about different functions.
Section 3: Introduction to Functions56
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Letβs Practice! 1. You earn $10.00 per hour babysitting. Your total earnings
depend on the number of hours you spend babysitting.
a. What is the independent variable?
b. What is the dependent variable?
c. How would you represent this situation using function notation?
2. The table below represents a relation.
ππ ππ a. Is the relation also a function? Justify your answer.
b. If the relation is not a function, what number could be changed to make it a function?
β3 5
0 4
2 6
β3 8
Try It! 3. Mrs. Krabappel is buying composition books for her
classroom. Each composition book costs $1.25.
a. What does her total cost depend upon?
b. What are the input and output?
c. Write a function to describe the situation.
d. If Mrs. Krabappel buys 24 composition books, they will cost her $30.00. Write this function using function notation.
Section 3: Introduction to Functions57
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4. Consider the following incomplete mapping diagrams.
a. Complete Diagram A so that it is a function.
b. Complete Diagram B so that it is NOT a function.
c. Is it possible to complete the mapping diagram for Diagram C so it represents a function? If so, complete the diagram to show a function. If not, justify your reasoning.
444
β137
Diagram A
444
β137
Diagram B
β137
Diagram C
444
BEAT THE TEST! 1. Isaac Messi is disorganized. To encourage Isaac to be
more organized, his father promised to give him three dollars for every day that his room is clean and his schoolwork is organized.
Part A: Define the input and output for the given
scenario.
Input:
Output:
Part B: Write a function to represent this situation.
Section 3: Introduction to Functions58
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2. The cost to manufacture π₯π₯ pairs of shoes can be represented by the function πΆπΆ π₯π₯ = 63π₯π₯. Complete the statement about the function.
If πΆπΆ(6) = 378, then pairs of shoes cost
3. Which of the following relations is not a function? A { 0, 5 , 2, 3 , 5, 8 , 3, 8 } B { 4, 2 , β4, 5 , 0, 0 } C { 6, 5 , 4, 1 , β3, 2 , 4, 2 } D {(β3, β3), (2, 1), (5, β2)}
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0663378
$6.$189.$378.$2,268.
Section 3 β Topic 2 Representing, Naming, and Evaluating Functions
A ball is thrown into the air with an initial velocity of 15 meters per second. The quadratic function β(π‘π‘) = β4.9π‘π‘> + 15π‘π‘ + 4 represents the height of the ball above the ground, in meters, with respect to time π‘π‘, in seconds. Determine β(2) and explain what it represents. Would β3 be a reasonable input for the function? The graph below represents the height of the ball with respect to time.
What is a reasonable domain for the function? What is a reasonable range for the function?
Time (in seconds)
Heig
ht (i
n m
eter
s)
Height of the Ball Over Time
Section 3: Introduction to Functions59
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Letβs Practice! 1. On the moon, the time, in seconds, it takes for an object
to fall a distance, ππ, in feet, is given by the function ππ ππ = 1.11 ππ.
a. Determine ππ(5) and explain what it represents.
b. The South Pole-Aitken basin on the moon is 42,768feet deep. Determine a reasonable domain for a rock dropped from the rim of the basin.
2. Floyd drinks two Mountain Dew sodas in the morning. The
function that represents the amount of caffeine, in milligrams, remaining in his body after drinking the sodas is given by ππ π‘π‘ = 110 0.8855 A where π‘π‘ is time in hours. Floyd says that in two days the caffeine is completely out of his system. Do you agree? Justify your answer.
Try It! 3. Medical professionals say that 98.6βis the normal body
temperature of an average person. Healthy individualsβ temperatures should not vary more than 0.5β from that temperature.
a. Write an absolute value function ππ(π‘π‘) to describe an
individualβs variance from normal body temperature, where π‘π‘ is the individualβs current temperature.
b. Determine ππ(101.5) and describe what that tells you about the individual.
c. What is a reasonable domain for a healthy individual?
Section 3: Introduction to Functions60
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BEAT THE TEST! 1. The length of a shipping box is two inches longer than the
width and four times the height.
Part A: Write a function ππ(π€π€) that models the volume of the box, where π€π€ is the width, in inches.
Part B: Evaluate ππ(10). Describe what this tells you about
the box.
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Section 3 β Topic 3 Adding and Subtracting Functions
Let β π₯π₯ = 2π₯π₯> + π₯π₯ β 5 and ππ π₯π₯ = β3π₯π₯> + 4π₯π₯ + 1. Find β π₯π₯ + ππ(π₯π₯). Find β π₯π₯ β ππ π₯π₯ .
Section 3: Introduction to Functions61
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Letβs Practice! 1. Consider the following functions.
ππ π₯π₯ = 3π₯π₯> + π₯π₯ + 2 ππ π₯π₯ = 4π₯π₯> + 2 3π₯π₯ β 4
β π₯π₯ = 5(π₯π₯> β 1)
a. Find ππ π₯π₯ β ππ π₯π₯ .
b. Find ππ π₯π₯ β β π₯π₯ .
Try It! 2. Recall the functions we used earlier.
ππ π₯π₯ = 3π₯π₯> + π₯π₯ + 2
ππ π₯π₯ = 4π₯π₯> + 2 3π₯π₯ β 4 β π₯π₯ = 5(π₯π₯> β 1)
a. Let ππ(π₯π₯) be ππ π₯π₯ + ππ π₯π₯ . Find ππ(π₯π₯).
b. Find β π₯π₯ βππ π₯π₯ .
Section 3: Introduction to Functions62
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BEAT THE TEST! 1. Consider the functions below.
ππ π₯π₯ = 2π₯π₯> + 3π₯π₯ β 5 ππ π₯π₯ = 5π₯π₯> + 4π₯π₯ β 1
Which of the following is the resulting polynomial when ππ π₯π₯ is subtracted from ππ(π₯π₯)? A β3π₯π₯> β π₯π₯ β 4
B β3π₯π₯> + 7π₯π₯ β 6
C 3π₯π₯> + π₯π₯ + 4
D 3π₯π₯> + 7π₯π₯ β 6
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Section 3 β Topic 4 Multiplying Functions
Use the distributive property and modeling to perform the following function operations. Let ππ π₯π₯ = 3π₯π₯> + 4π₯π₯ + 2 and ππ π₯π₯ = 2π₯π₯ + 3. Find ππ(π₯π₯) β ππ(π₯π₯).
Section 3: Introduction to Functions63
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Let ππ π¦π¦ = 3π¦π¦G β 2π¦π¦> + 8 and ππ π¦π¦ = π¦π¦> β 2. Find ππ(π¦π¦) β ππ(π¦π¦).
Letβs Practice! 1. Let β π₯π₯ = π₯π₯ β 1 and ππ π₯π₯ = π₯π₯I + 6π₯π₯> β 5.
Find β(π₯π₯) β ππ(π₯π₯).
Section 3: Introduction to Functions64
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Try It! 2. The envelope below has a mailing label.
a. Let π΄π΄ π₯π₯ = πΏπΏ π₯π₯ β ππ π₯π₯ βππ(π₯π₯) β ππ(π₯π₯). Find π΄π΄(π₯π₯).
πΏπΏ(π₯π₯) = 6π₯π₯ + 5
MR. AL GEBRA 123 INFINITY WAY POLYNOMIAL, XY 11235
ππ(π₯π₯) = 6π₯π₯ + 5 ππ(π₯π₯) = π₯π₯ + 4
ππ( π₯π₯)=π₯π₯+2
b. What does the function π΄π΄(π₯π₯) represent in this problem?
Section 3: Introduction to Functions65
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BEAT THE TEST! 1. The length of the sides of a square are π π inches long.
A rectangle is six inches shorter and eight inches wider than the square. Part A: Express both the length and the width of the
rectangle as a function of a side of the square. Part B: Write a function to represent the area of the
rectangle in terms of the sides of the square.
2. Felicia needs to find the area of a rectangular field in her backyard. The length is represented by the function πΏπΏ π₯π₯ = 4π₯π₯P β 3π₯π₯> + 6 and the width is represented by the functionππ π₯π₯ = π₯π₯ + 1. Which of the following statements is correct about the area, π΄π΄ π₯π₯ ,of the rectangular field in Feliciaβs backyard? Select all that apply. Β¨ π΄π΄ π₯π₯ = 2[πΏπΏ π₯π₯ +ππ π₯π₯ ] Β¨ The resulting expression for π΄π΄(π₯π₯) is a fifth-degree
polynomial. Β¨ The resulting expression for π΄π΄(π₯π₯) is a polynomial with a
leading coefficient of 5. Β¨ The resulting expression for π΄π΄(π₯π₯) is a binomial with a
constant of 6. Β¨ ππ π₯π₯ = S(T)
U(T)
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Section 3: Introduction to Functions66
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Section 3 β Topic 5 Closure Property
When we add two integers, what type of number is the sum? When we multiply two irrational numbers, what type of numbers could the resulting product be? A set is ___________ for a specific operation if and only if the operation on two elements of the set always produces an element of the same set. Are integers closed under addition? Justify your answer. Are irrational numbers closed under multiplication? Justify your answer. Are integers closed under division? Justify your answer.
Letβs apply the closure property to polynomials. Are the following statements true or false? If false, give a counterexample. Polynomials are closed under addition. Polynomials are closed under subtraction. Polynomials are closed under multiplication. Polynomials are closed under division.
Section 3: Introduction to Functions67
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Letβs Practice! 1. Check the boxes for the following sets that are closed
under the given operations.
Set + β Γ Γ·
{0, 1, 2, 3, 4, β¦ } Β¨ Β¨ Β¨ Β¨
{β¦ ,β4,β3,β2,β1} Β¨ Β¨ Β¨ Β¨
{β¦ ,β3,β2,β1, 0, 1, 2, 3, β¦ } Β¨ Β¨ Β¨ Β¨
{rational numbers} Β¨ Β¨ Β¨ Β¨
{polynomials} Β¨ Β¨ Β¨ Β¨
Try It! 2. Ms. Sanabria claims that the closure properties for
polynomials are analogous to the closure properties for integers. Mr. Roberts claims that the closure properties for polynomials are analogous to the closure properties for rational numbers. Who is correct? Explain your answer.
Section 3: Introduction to Functions68
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BEAT THE TEST! 1. Choose from the following words and expressions to
complete the statement below.
5π¦π¦YZ + 7π₯π₯> + 8π¦π¦>
The product of 5π₯π₯P β 3π₯π₯> + 2 and _______________________
illustrates the closure property because the
_______________ of the product are ____________________ ,
and the product is a polynomial.
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2π₯π₯G + (3π¦π¦)Y> β 2 (5π¦π¦)> + 4π₯π₯ + 3π¦π¦I
integers variables whole numbers coefficients rational exponents
numbers
Section 3 β Topic 6 Real-World Combinations and Compositions of
Functions There are many times in real world situations when we must combine functions. Profit and revenue functions are a great example of this. Letβs Practice! 1. At the fall festival, the senior class sponsors hayrides to
raise money for the senior trip. The ticket price is $5.00 and each hayride carries an average of 15people. They consider raising the ticket price in order to earn more money. For each $0.50 increase in price, an average of 2 fewer seats will be sold. Let π₯π₯ represent the number of $0.50increases.
a. Write a function, ππ(π₯π₯), to represent the cost of one
ticket based on the number of increases.
b. Write a function, π π (π₯π₯), to represent the number of riders based on the number of increases.
c. Write a revenue function for the hayride that could be used to maximize revenue.
Section 3: Introduction to Functions69
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Try It!
2. The freshman class is selling t-shirts to raise money for a field trip. The cost of each custom designed t-shirt is $8. There is a $45 fee to create the design. The class plans to sell the shirts for $12.
a. Define the variable.
b. Write a cost function.
c. Write a revenue function.
d. Write a profit function.
Letβs Practice! 3. Priscilla works at a cosmetics store. She receives a weekly
salary of $350 and is paid a 3% commission on weekly sales over $1500.
a. Let π₯π₯ represent Priscillaβs weekly sales. Write a function,
ππ π₯π₯ , to represent Priscillaβs weekly sales over $1500.
b. Let π₯π₯ represent the weekly sales on which Priscilla earns commission. Write a function, ππ π₯π₯ , to represent Priscillaβs commission.
c. Write a composite function, (ππ β ππ)(π₯π₯) to represent the amount of money Priscilla earns on commission.
Section 3: Introduction to Functions70
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Try It! 4. A landscaping company installed a sprinkler that rotates
and sprays water in a circular pattern. The water reaches its maximum radius of 10feet after 30 seconds. The company wants to know the area that the sprinkler is covering at any given time after the sprinkler is turned on.
a. Let π‘π‘ represent the time in seconds after the sprinkler is
turned on. Write a function, ππ(π‘π‘), to represent the size of the growing radius based on time after the sprinkler is turned on.
b. Let ππ represent the size of the radius at any given time. Write a function, π΄π΄(ππ), to represent the area that the sprinkler covers at any given time, in seconds.
c. Write a composite function, π΄π΄ ππ(π‘π‘) to represent the area based on the time, in seconds, after the sprinkler is turned on.
BEAT THE TEST! 1. A furniture store charges 6.5% sales tax on the cost of the
furniture and a $20 delivery fee. (The delivery fee is not subject to sales tax.)
The following functions represent the situation:
ππ ππ = 1.065ππ ππ ππ = ππ + 20
Part A: Write the function ππ(ππ ππ ).
Part B: Match each of the following to what they represent. Some letters will be used twice.
ππ A. The cost of the furniture,
sales tax, and delivery fee. ππ B. The cost of the furniture and
sales tax. ππ ππ C. The cost of the furniture.
ππ(ππ)
ππ(ππ ππ )
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Section 3: Introduction to Functions71
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Section 3 β Topic 7 Key Features of Graphs of Functions β Part 1
Letβs review the definition of a function. Every input value (π₯π₯) corresponds to ___________ _______ output value π¦π¦ . Consider the following graph. How can a vertical line help us quickly determine if a graph represents a function? We call this the vertical line test. Use the vertical line test to determine if the graph above represents a function.
Important facts:
Γ Graphs of lines are not always functions. Can you describe a graph of a line that is not a function?
Γ Functions are not always linear.
Sketch a graph of a function that is not linear.
Section 3: Introduction to Functions72
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3. Consider the following scenarios. Determine if each one represents a function or not. a. An analyst takes a survey of people about their
heights (in inches) and their ages. She then relates their heights to their ages (in years).
b. A geometry student is dilating a circle and analyzes the area of the circle as it relates to the radius.
c. A teacher has a roster of 32 students and relates the studentsβ letter grades to the percent of points earned.
d. A boy throws a tennis ball in the air and lets it fall to the ground. The boy relates the time passed to the height of the ball.
Letβs Practice! 1. Use the vertical line test to determine if the following
graphs are functions.
Try It! 2. Which of the following graphs represent functions? Select
all that apply.
Β¨ Β¨ Β¨ Β¨
Β¨ Β¨ Β¨ Β¨
Section 3: Introduction to Functions73
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Itβs important to understand key features of graphs.
Γ An ππ-intercept of a graph is the location where the graph crosses the _____________.
Γ The π¦π¦-coordinate of the π₯π₯-intercept is always _______. Γ The ππ-intercept of a graph is the location where the
graph crosses the _____________. Γ The π₯π₯-coordinate of the π¦π¦-intercept is always
__________. Γ The π₯π₯-intercept is the _____________ to a function or
graph. All of these features are very helpful in understanding real-world context.
Letβs Practice! 4. Consider the following graph that represents the height, in
feet, of a water balloon dropped from a 2nd story window after a given number of seconds.
a. What is the π₯π₯-intercept? b. What is the π¦π¦-intercept? c. Label the intercepts on the graph.
Section 3: Introduction to Functions74
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Section 3 β Topic 8 Key Features of Graphs of Functions β Part 2
Letβs discuss other key features of graphs of functions. Γ Domain: the input or the ______ values. Γ Range: the ____________________ or the π¦π¦-values.
Γ Increasing intervals: as the π₯π₯-values _________________,
the π¦π¦-values _________________.
Γ Decreasing intervals: as the π₯π₯-values _________________, the π¦π¦-values _________________.
Γ Relative maximum: the point on a graph where the
interval changes from __________________ to __________________.
Γ Relative minimum: the point on a graph where the
interval changes from __________________ to __________________.
We read a graph from left to right to determine if it is increasing or decreasing, like reading a book.
Try It! 5. Refer to the previous problem for the following questions.
a. What does the π¦π¦-intercept represent in this real-world context?
b. What does the π₯π₯-intercept represent in this real-world
context? c. What is the solution to this situation?
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Section 3: Introduction to Functions75
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Letβs Practice! 1. Use the following graph of an absolute value function to
answer the questions below.
a. Define the domain.
b. Define the range.
c. Where is the graph increasing?
d. Where is the graph decreasing?
e. Identify any relative maximums.
f. Identify any relative minimums.
Try It! 2. Use the graph of the following quadratic function to
answer the questions below.
a. Define the domain.
b. Define the range.
c. Where is the graph increasing?
d. Where is the graph decreasing?
e. Identify any relative maximums. f. Identify any relative minimums.
Section 3: Introduction to Functions76
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BEAT THE TEST! 1. The following graph is a piecewise function.
Which of the following statements are true about the graph? Select all that apply. Β¨ The graph is increasing when the domain is
β6 < π₯π₯ < β4. Β¨ The graph has exactly one relative minimum. Β¨ The graph is increasing when β4 β€ π₯π₯ β€ 0. Β¨ The graph is increasing when π₯π₯ > 4. Β¨ The graph is decreasing when the domain is
π₯π₯ π₯π₯ < β6 βͺ π₯π₯ > 2 . Β¨ The range is π¦π¦ 0 β€ π¦π¦ < 4 βͺ π¦π¦ β₯ 5 . Β¨ There is a relative minimum at (2, 2).
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3. Describe everything you know about the key features of the following graph of an exponential function.
Section 3: Introduction to Functions77
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Section 3 β Topic 9 Average Rate of Change Over an Interval
Consider the following graph of the square root function ππ π₯π₯ = π₯π₯ .
Draw a line connecting ππ and ππ. Determine the slope of the line between the interval ππ, ππ . For every two points π₯π₯Z and π₯π₯>, where π₯π₯Z β π₯π₯>, π₯π₯Z, π¦π¦Z and π₯π₯>, π¦π¦> form a straight line and create a _________ ___________.
To determine the average rate of change for any function ππ π₯π₯ over an interval, we can use two points (π₯π₯Z, _______) and (π₯π₯>, _______)that lie on that interval. The process to find the slope of a linear function is:
ππ =π¦π¦> β π¦π¦Zπ₯π₯> β π₯π₯Z
We can also use the slope formula to find the average rate of change over an interval [ππ, ππ], where π₯π₯Z = ππ and π₯π₯> = ππ.
Letβs Practice! 1. Tom is jumping off the diving board at the Tony Dapolito
Pool. His height is modeled by the quadratic function β π‘π‘ = βπ‘π‘> + 2π‘π‘ + 4, where β π‘π‘ represents height above water (in feet), and π‘π‘ represents time after jumping (in seconds).
a. Determine the average rate of change for the following intervals. ππ, ππ
ππ, ππ
ππ, ππ
b. Compare Tomβs average rate of change over the
interval ππ, ππ with his average rate of change over the interval [ππ, ππ]. What does this represent in real life?
Section 3: Introduction to Functions78
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3. Determine the intervals that have the same average rate of change in the graph ππ π₯π₯ = π₯π₯l below.
Try It! 2. Consider the table for the exponential function, ππ π₯π₯ = 3T,
shown below.
Point ππ ππ ππ
ππ 0 1
ππ 1 3
π π 2 9
ππ 3 27
a. Determine the average rate of change over the
interval [ππ, ππ]. b. Compare the average rate of change over the
interval ππ,ππ with the average rate of change over the interval [π π , ππ].
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BEAT THE TEST!
1. Suppose that the cost of producing ππ radios is defined by ππ ππ = 300 + 15ππ β 0.3ππ>. Determine which of the following intervals has the greatest average rate of change for the cost to produce a radio.
A Between 20 and 25 radios. B Between 60 and 65 radios. C Between 5 and 10 radios. D Between 30 and 35 radios.
2. Consider the absolute value function ππ(π₯π₯)and the step function ππ π₯π₯ in the graphs below.
ππ π₯π₯ ππ(π₯π₯)
Which of the following is true about the rate of change of the graphs? A The average rate of change for ππ π₯π₯ over the interval
ππ, ππ is greater than the average rate of change for ππ(π₯π₯) over the interval [ππ, ππ].
B The average rate of change for ππ(π₯π₯) over the interval [ππ, ππ] is greater than the average rate of change for ππ(π₯π₯)over the interval[ππ, ππ].
C The average rate of change for ππ π₯π₯ over the interval [ππ, ππ] is βZ
>.
D The average rate of change for ππ(π₯π₯) over the interval [ππ, ππ] is βZ
>.
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Section 3: Introduction to Functions80
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The following graphs are transformations of ππ(π₯π₯). Describe what happened in each graph.
ππ π₯π₯ + 2
ππ π₯π₯ β 1
ππ π₯π₯ + 2
ππ π₯π₯ β 1
Which graphs transformed the independent variable? Which graphs transformed the dependent variable?
Section 3 β Topic 10 Transformations of Functions
The graph of ππ(π₯π₯) is shown below.
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Letβs Practice! 1. For the following functions, state whether the independent
or dependent variable is being transformed and describe the transformation (assume ππ > 0).
a. ππ π₯π₯ + ππ
b. ππ π₯π₯ β ππ
c. ππ(π₯π₯ + ππ)
d. ππ(π₯π₯ β ππ)
2. The following table represents the function ππ π₯π₯ .
ππ ππ(ππ)
β2 0.25
β1 0.5
0 1
1 2
2 4
The function β π₯π₯ = ππ 2π₯π₯ . Complete the table for β π₯π₯ .
ππ ππ(ππππ) ππ(ππ)
β1 ππ(2(β1))
β0.5 ππ(2(β0.5))
0
0.5
1
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4. The graph ofππ(π₯π₯) is shown below.
Let ππ π₯π₯ = ππ π₯π₯ + 3 β 2. Graph ππ(π₯π₯) on the coordinate plane with ππ π₯π₯ .
Try It! 3. The table below shows the values for the function ππ π₯π₯ .
ππ β2 β1 0 1 2 ππ(ππ) 4 2 0 2 4
Complete the table for the function βZ>ππ(π₯π₯).
ππ βππππππ(ππ)
β2
β1
0
1
2
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BEAT THE TEST! 1. The graph of ππ(π₯π₯) is shown below.
Let ππ π₯π₯ = ππ(π₯π₯ β 3) and β π₯π₯ = ππ π₯π₯ β 3. Graph ππ(π₯π₯) and β(π₯π₯) on the coordinate plane with ππ π₯π₯ .
2. The table below shows the values for the function ππ(π₯π₯).
ππ β4 β1 0 2 3 ππ(ππ) 12 6 4 8 10
Complete the table for the function Z
>ππ π₯π₯ β 3.
ππ ππππππ ππ β ππ
Test Yourself! Practice Tool
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