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295A
4.1Planting the SeedsExploring Cubic Functions
TEXAS ESSENTIAL KNOWLEDGE
AND SKILLS FOR MATHEMATICS
(2) Attributes of functions and their inverses.
The student applies mathematical processes
to understand that functions have distinct key
attributes and understand the relationship
between a function and its inverse. The student
is expected to:
(A) graph the functions f(x) 5 √__
x , f(x) 5 1 __ x ,
f(x) 5 x 3 , f(x) 5 3 Ï·· x , f(x) 5 b x , f(x) 5 |x|,
and f(x) 5 logb(x) where b is 2, 10, and
e, and, when applicable, analyze the
key attributes such as domain, range,
intercepts, symmetries, asymptotic
behavior, and maximum and minimum
given an interval
ESSENTIAL IDEAS
A cubic function is a function that can be written in the standard form f(x) = a x 3 1 b x 2 1 cx 1 d where a fi 0.
Multiple representations such as tables, graphs, and equations are used to represent cubic functions.
A relative maximum is the highest point in a particular section of a graph.
A relative minimum is the lowest point in a particular section of a graph.
Key characteristics are used to interpret the graph of cubic functions.
Characteristics and behaviors of cubic functions are related to its factors.
Multiplicity is how many times a particular number is a zero for a given polynomial function.
The Fundamental Theorem states that a cubic function must have 3 roots.
LEARNING GOALS KEY TERMS
relative maximum
relative minimum
cubic function
multiplicity
In this lesson, you will:
Represent cubic functions using words, tables, equations, and graphs.
Interpret the key characteristics of the graphs of cubic functions.
Analyze cubic functions in terms of their mathematical context and problem context.
Connect the characteristics and behaviors of cubic functions to its factors.
Compare cubic functions with linear and quadratic functions.
Build cubic functions from linear and quadratic functions.
295B Chapter 4 Polynomial Functions
4
(6) Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student
applies mathematical processes to understand that cubic, cube root, absolute value and rational
functions, equations, and inequalities can be used to model situations, solve problems, and make
predictions. The student is expected to:
(A) analyze the effect on the graphs of f(x) 5 x 3 and f(x) 5 3 Ï·· x when f(x) is replaced by af(x), f(bx),
f(x 2 c), and f(x) 1 d for speci!c positive and negative real values of a, b, c, and d
(7) Number and algebraic methods. The student applies mathematical processes to simplify and
perform operations on expressions and to solve equations. The student is expected to:
(I) write the domain and range of a function in interval notation, inequalities, and set notation
Overview
The terms cubic function, relative minimum, relative maximum, and multiplicity are de!ned in this lesson.
The standard form of a cubic equation is given. In the !rst activity, a rectangular sheet of copper is used
to create planters if squares are removed from each corner of the sheet and the sides are then folded
upward. Students will analyze several sized planters and calculate the volume of each size. They then
write a volume function in terms of the height, length, and width and graph the function using a graphing
calculator. Using key characteristics, students analyze the graph and conclude that the graph is cubic.
Students differentiate the domain and range of the problem situation from the domain and range of the
cubic function. Using a graphing calculator and speci!ed volumes, students determine which if any
possible sized of planters meet the criteria. The second activity is similar but uses a cylindrical planter.
The volume function is written in three forms and students will algebraically and graphically verify their
equivalence. A graphing calculator is used throughout this lesson.
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4.1 Exploring Cubic Functions 295C
Warm Up
Simplify each expression and identify its function family.
1. (x 1 4) (10)
(x 1 4) (10) 5 10x 1 40
Linear Function
2. (x 2 4) (x 2 5)
(x 2 4) (x 2 5) 5 x 2 2 9x 1 20
Quadratic Function
3. (x 1 8) 2
(x 1 8) 2 5 x 2 1 16x 1 64
Quadratic Function
4. (x 2 4) (x 2 5) (x 2 1)
(x 2 4) (x 2 5) 5 x 2 + 9x 1 20
( x 2 1 9x 1 20) (x 2 1) 5 x 3 1 8x 2 1 11x 2 20
Cubic Function
295D Chapter 4 Polynomial Functions
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295
LEARNING GOALS
4.1
If you have ever been to a 3D movie, you know that it can be quite an interesting
experience. Special film technology and wearing funny-looking glasses allow movie-
goers to see a third dimension on the screen—depth. Three dimensional filmmaking
dates as far back as the 1920s. As long as there have been movies, it seems that
people have looked for ways to transform the visual experience into three dimensions.
However, your brain doesn’t really need special technology or silly glasses to
experience depth. Think about television, paintings, and photography—artists have
been making two-dimensional works of art appear as three-dimensional for a long
time. Several techniques help the brain perceive depth. An object that is closer is
drawn larger than a similarly sized object off in the distance. Similarly, an object in
the foreground may be clear and crisp while objects in the background may appear
blurry. These techniques subconsciously allow your brain to process depth in
two dimensions.
Can you think of other techniques artists use to give the illusion of depth?
KEY TERMS
relative maximum
relative minimum
cubic function
multiplicity
In this lesson, you will:
Represent cubic functions using words,
tables, equations, and graphs.
Interpret the key characteristics of the
graphs of cubic functions.
Analyze cubic functions in terms of their
mathematical context and problem context.
Connect the characteristics and behaviors
of cubic functions to its factors.
Compare cubic functions with linear and
quadratic functions.
Build cubic functions from linear and
quadratic functions.
Planting the SeedsExploring Cubic Functions
296 Chapter 4 Polynomial Functions
4
Problem 1
A planter is constructed from
a rectangular sheet of copper.
Given the dimensions of the
rectangular sheet, students
will complete a table of values
listing various sizes of planters
with respect to the length
width, height, and volume. They
describe observable patterns,
analyze the relationship
between the height, length,
and width, and write a function
to represent the volume of
the planter box. Students
use a graphing calculator to
graph the function, describe
the key characteristics of the
graph, identify the maximum
volume, the domain and range
of the function versus the
problem situation and conclude
that it is cubic. The terms
relative maximum and relative
minimum are de!ned. Given
speci!ed volumes, students
use a graphing calculator
to determine possible sized
planter boxes that meet
the criteria.
Grouping
Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Question 1, part (a)
How did you determine the
height of each planter in the
table of values?
How did you determine the width of each planter in the table of values?
How did you determine the length of each planter in the table of values?
How did you determine the volume of each planter in the table of values?
How did you determine each of the expressions when the length of the corner
side was h inches in the table of values?
PROBLEM 1 Business Is Growing
The Plant-A-Seed Planter Company produces planter boxes. To make the boxes, a square is
cut from each corner of a rectangular copper sheet. The sides are bent to form a rectangular
prism without a top. Cutting different sized squares from the corners results in different
sized planter boxes. Plant-A-Seed takes sales orders from customers who request a sized
planter box.
Each rectangular copper sheet is 12 inches
by 18 inches. In the diagram, the solid lines
indicate where the square corners are cut and
the dotted lines represent where the sides
are bent for each planter box.
h
18 inches
12 inches
h
hhh
h
h
h
1. Organize the information about each sized planter box made from a 12 inch by 18 inch
copper sheet.
a. Complete the table. Include an expression for each planter box’s height, width,
length, and volume for a square corner side of length h.
Square
Corner Side
Length
(inches)
Height
(inches)
Width
(inches)
Length
(inches)
Volume
(cubic inches)
0 0 12 18 0
1 1 10 16 160
2 2 8 14 224
3 3 6 12 216
4 4 4 10 160
5 5 2 8 80
6 6 0 6 0
7 7 22 4 256
h h 12 2 2h 18 2 2h h(12 2 2h)(18 2 2h)
Recall the volume
formula V 5 lwh.
It may help to create a model of the planter
by cutting squares out of the corners of a sheet of paper and folding.
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4.1 Exploring Cubic Functions 297
Guiding Questions for Share Phase, Question 1, part (b) and Question 2
Is the height the same as the
side length of each square?
How does the increasing
height affect the width and
length of planter?
As the height increases by
one inch, what happens to
the width and the length?
If the volume is 0 cubic
inches, what does that
mean with respect to the
problem situation?
What happens to the width of
the planter if the size of the
square corner is equal to
6 inches?
Is the corner square’s length
subtracted from the length
and width of the planter box?
How was the table of values
useful when writing the
function for the volume of the
planter box?
b. What patterns do you notice in the table?
The height is the same as the side length of the square.
As the height increases, the width and length decrease.
For every inch increase in height, the width and length decrease by 2 inches.
The volume starts at 0 cubic inches, increases, and then decreases back to
0 cubic inches.
2. Analyze the relationship between the height, length, and width of each planter box.
a. What is the largest sized square corner that can be cut to make a planter box?
Explain your reasoning.
The size of the square corner must be less than 6 inches. A 6-inch square would
result in a width of 0 inches.
b. What is the relationship between the size of the corner square and the length and
width of each planter box?
Twice the corner square’s length is subtracted from the length and the width of
each planter box.
For example, a 1-inch cut corner square results in a length of 18 2 (2 3 1) and a
width of 12 2 (2 3 1).
c. Write a function V(h) to represent the volume of the planter box in terms of the
corner side of length h.
V(h) 5 h(12 2 2h)(18 2 2h)
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Grouping
Have students complete
Questions 3 and 4 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Question 3
What is a complete graph?
Did Louis, Ahmed, or Heidi
sketch a complete graph?
Does Louis’s graph have an
axis of symmetry?
What is the interval in which
the graph increases?
What is the interval in which
the graph decreases?
? 3. Louis, Ahmed, and Heidi each used a graphing calculator to analyze the volume function,
V(h), and sketched their viewing window. They disagree about the shape of the graph.
Louis
x
y
height
volume
The graph increases and then
decreases. It is a parabola.
Ahmed
x
y
height
volu
me
The graph lacks a line of
symmetry, so it can’t be a
parabola.
Heidi
x
y
height
volu
me
I noticed the graph curves back up so it can’t be a parabola.
Evaluate each student’s sketch and rationale to determine who is correct.
For the student(s) who is/are not correct, explain why the rationale is not correct.
Ahmed and Heidi are correct.
Louis is not correct. The graph is not a parabola because it does not have a line of
symmetry. Extending the viewing window on the graph also shows that the graph
curves back up.
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Guiding Questions for Share Phase, Question 4
Where are the x-intercepts?
Where are the y-intercepts?
Does the function have
a maximum or a
minimum point?
Where on the graph is the
point which describes the
maximum volume of
a planter box?
What is the signi!cance of
the x-value at the maximum
point of the function?
What is the signi!cance of
the y-value at the maximum
point of the function?
Why is the domain of the
function different than the
domain of the problem
situation?
Why is the range of the
function different than
the domain of the
problem situation?
Does it make sense to have a
planter box with the height of
0 inches?
4. Represent the function on a graphing calculator using the window
[210, 15] 3 [2400, 400].
a. Describe the key characteristics of the graph?
The graph increases until it reaches a peak and then decreases.
The x-intercepts are (0, 0) and (6, 0) and (9, 0).
The y-intercept is (0, 0).
The graph first increases and then begins to
decrease at (2.35, 228). Then the graph
continues to decrease and finally begins to
increase at (7.65, 268.16).
b. What is the maximum volume of a planter box?
State the dimensions of this planter box.
Explain your reasoning.
The maximum volume is 228 cubic inches.
The dimensions of this planter are 2.35 inches 3
7.30 inches 3 13.30 inches.
Graphically this is the highest point between
x 5 0 and x 5 6.
c. Identify the domain of the function V(h).
Is the domain the same or different in terms of the context of this problem?
Explain your reasoning.
The domain of the function is (2`, `).
In terms of this problem situation, only the height values (0, 6) make sense. Values
outside of this domain result in negative planter box dimensions.
d. Identify the range of the function V(h).
Is the range the same or different in terms of the context of this problem?
Explain your reasoning.
The range of the function is (2`, `).
The range in terms of this problem situation is (0, 228) because the maximum
volume is 228 and it is impossible to have a volume less than 0.
e. What do the x-intercepts represent in this problem situation? Do these values make
sense in terms of this problem situation? Explain your reasoning.
The x-intercepts represent the planter box heights in which the volume is 0 cubic
inches. It does not make sense to have a planter box with a height of 0 inches.
In this problem you are
determining the maximum value graphically, but consider
other representations. How will your solution strategy change when
using the table or equation?
300 Chapter 4 Polynomial Functions
4
The key characteristics of a function may be different within a given domain. The function
V(h) 5 h(12 2 2h)(18 2 2h) has x-intercepts at x 5 0, x 5 6, and x 5 9.
x0
100
2 4 6 8
y
V(h)
(2.35, 228)200
2100
2200Volume (cubic inches)
Height (inches)
As the input values for height increase, the output values for volume approach in!nity.
Therefore, the function doesn’t have a maximum; however, the point (2.35, 228) is a relative
maximum within the domain interval of (0, 6). A relative maximum is the highest point in a
particular section of a graph. Similarly, as the values for height decrease, the output values
approach negative in!nity. Therefore, a relative minimum occurs at (7.65, 268.16). A relative
minimum is the lowest point in a particular section of a graph.
The function v(h) represents all of the possible volumes for a given height h. A horizontal line
is a powerful tool for working backwards to determine the possible values for height when
the volume is known.
The given volume of a planter box is 100 cubic inches. You can determine the
possible heights from the graph of V(h).
x0 2
Volume (cubic inches)
Height (inches)
4 6 8
y
200
2100
2200
100V(h)
y = 100
Draw a horizontal line at y 5 100.
Identify each point where V(h) intersects
with y 5 100, or where V(h) 5 100.
The !rst point of intersection is represented using function notation as V(0.54) 5 100.
Grouping
Ask a student to read the
information and discuss the
worked example as a class.
Complete Question 5
as a class.
Guiding Questions for Discuss Phase
What is the difference
between the maximum point
on the graph of a function
and a relative maximum point
on the graph of a function?
What is the difference
between the minimum point
on the graph of a function
and a relative minimum point
on the graph of a function?
Can the width or the length
of the planter box have a
negative value?
Which values on the volume
function result in negative
values for the width or length
of the planter box?
If a horizontal line such
as y 5 50 is graphed with
the volume function on a
coordinate plane, what is the
signi!cance of the points
of intersection?
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4.1 Exploring Cubic Functions 301
5. A customer ordered a particular planter box with a volume of 100 cubic inches, but did
not specify the height of the planter box.
a. Use a graphing calculator to determine when V(h) 5 100. Then write the intersection
points in function notation. What do the intersection points mean in terms of this
problem situation?
V(0.54) 5 100,
V(4.76) 5 100,
V(9.70) 5 100,
The intersection points are the heights that create a planter box with a volume of
100 cubic inches.
b. How many different sized planter boxes can Plant-A-Seed make to !ll this order?
Explain your reasoning.
The graph has 3 solutions, the points of intersection (0.54, 100), (4.76, 100), and
(9.70, 100).
The first two intersection points lie within the domain of this problem context,
indicating that planter boxes with height 4.76 inches and 0.54 inches have a
volume of 100 cubic inches.
A height of 9.70 inches also results in a volume of 100 cubic inches, but this value
does not make sense in this problem situation. This height is not within the
domain because it leads to negative values for length and width.
6. A neighborhood beautifying committee would like to purchase a variety of planter boxes
with volumes of 175 cubic inches to add to business window sill store fronts. Determine
the planter box dimensions that the Plant-A-Seed Company can create for the
committee. Show all work and explain your reasoning.
The 2 planters with dimensions h 5 3.78, l 5 10.44, w 5 4.44 and h 5 1.15, l 5 15.70,
w 5 9.70 have a volume of 175 cubic inches.
The function has 3 graphical solutions, but only 2 possible planters make sense in
this problem situation.
I graphed the volume function and the horizontal line y 5 175. The intersection points
are the solutions to this problem.
Guiding Questions for Share Phase, Question 5
How did you identify the
points at which v(h) 5 100
intersected the
volume function?
What values are represented
on the y-axis with respect to
the problem situation? What
is the unit of measure?
What values are represented
on the x-axis with respect to
the problem situation? What
is the unit of measure?
Are all three solutions
reasonable with respect to
the problem situation? Why
or why not?
Grouping
Have students complete
Questions 6 through 8 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Question 6
How is this problem situation
different than the last
problem situation?
How many times does
the horizontal line
y 5 175 intersect the
volume function?
Are all three points of
intersection relevant to the
problem situation? Why or
why not?
302 Chapter 4 Polynomial Functions
4
Guiding Questions for Share Phase, Questions 7 and 8
If the area of the base of
the planter box is 12 square
inches, what does this tell
you about the length and
width of the planter box?
What algebraic expressions
are used to determine the
length and width of the
planter box?
What equation can be used
to determine the length and
width of the planter box?
When the equation
representing the area of the
planter box is graphed, what
is represented on the x-
and y-axis?
What is the width of a planter
box that has a height of
5 inches?
What is the length of a
planter box that has a height
of 5 inches?
How many planter boxes
have a height of 5 inches?
What is the volume of a
planter box that has a height
of 5 inches?
7. Plant-A-Seed’s intern claims that he can no longer complete the order because he
spilled a cup of coffee on the sales ticket. Help Jack complete the order by determining
the missing dimensions from the information that is still visible. Explain how you
determined possible unknown dimensions of each planter box.
Plant-A-Seed
Sales Ticket
Base Area: 12 square inchesHeight:Length:Width:Volume:
The height of the planter box is 5.21 inches, the length is 7.58 inches, and the width
is 1.58 inches.
I set up the equation (18 2 2x)(12 2 2x) 5 12. Then I used a graphing calculator to
graph y1 5 (18 2 2x)(12 2 2x) and y
2 5 12 and calculated the intersection points. The
intersection points are x 5 5.21 and x 5 9.79. However, the second value is greater
than 6, so it doesn’t make sense in this problem situation.
Finally, I substituted x 5 5.21 back into the expression (18 2 2x)(12 2 2x) to
determine the values for the length and width of the planter box.
8. A customer sent the following email:
To Whom It May Concern,
I would like to purchase several planter boxes, all with a height
of 5 inches. Can you make one that holds 100 cubic inches
of dirt? Please contact me at your earliest convenience.
Thank you,
Muriel Jenkins
Write a response to this customer, showing all calculations.
Dear Ms. Jenkins,
Unfortunately, the Plant-A-Seed Company cannot create a
planter box that holds 100 cubic inches of dirt and has a
height of 5 inches. The height of the planter box determines
the other dimensions. The three dimensions then determine the volume.
A planter box with a height of 5 inches is 2 inches wide and 8 inches long. It will hold
80 cubic inches of dirt. This is our only planter box available with a height of
5 inches.
I hope that this option for planter boxes will work for you.
Sincerely,
Plant-A-Seed
How is the volume
function built in this problem?
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Problem 3
The volume function from
Problem 1 is written in three
different forms; the product
of three linear functions, the
product of a linear function
and a quadratic function, and
a cubic function in standard
form. Students will algebraically
and graphically verify the three
forms of the volume function
are equivalent.
Grouping
Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses as
a class.
Guiding Questions for Share Phase, Question 1
What algebraic properties
were used to show the
functions were equivalent?
What should happen
graphically, if the three
functions are equivalent?
Do all three forms of the
function produce the
same graph?
PROBLEM 3 Cubic Equivalence
Let’s consider the volume formula from Problem 1, Business is Growing.
1. Three forms of the volume function V(h) are shown.
V(h) 5 h(18 2 2h)(12 2 2h) V(h) 5 h(4h2 2 60h 1 216) V(h) 5 4h3 2 60h2 1 216h
The product of three linear
functions that represent
height, length, and width.
The product of a linear
function that represent
the height and a quadratic
function representing the
area of the base.
A cubic function in
standard form.
a. Algebraically verify the functions are equivalent. Show all work and explain
your reasoning.
V(h) 5 h(18 2 2h)(12 2 2h)
5 h(216 2 36h 2 24h 1 4h2)
5 h(216 2 60h 1 4h2)
5 216h 2 60h2 1 4h3
5 4h3 2 60h2 1 216h
V(h) 5 h(4h2 2 60h 1 216)
5 4h3 2 60h2 1 216h
V(h) 5 4h3 2 60h2 1
216h
b. Graphically verify the functions are equivalent. Sketch all three functions and explain
your reasoning.
x
1000
2000
21000
20100220 210 4030240 230
y
22000
23000
24000
3000
4000
Graphing each of the functions on the same coordinate plane, I notice that they
all produce the same graph. This means that the functions must be equivalent.
c. Does the order in which you multiply factors matter? Explain your reasoning.
No. The order in which I multiply factors doesn’t matter. The properties of integers
hold for manipulating expressions algebraically. In this case, the Associative
Property of Multiplication holds true.
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308 Chapter 4 Polynomial Functions
4
You can determine the product of the linear factors (x 1 2)(3x 2 2)(4 1 x) using
multiplication tables.
Step 1: Step 2:
Choose 2 of the binomials to multiply. Multiply the product from step 1 with the
Then combine like terms. remaining binomial.
Then combine like terms.
? x 2
3x 3x2 6x
22 22x 24
? 4 x
3x2 12x2 3x3
4x 16x 4x2
24 216 24x
(x 1 2)(3x 2 2)(4 1 x) 5 3x3 1 16x2 1 12x 2 16.
2. Analyze the worked example for the multiplication of three binomials.
a. Use a graphing calculator to verify graphically that the expression in factored form is
equivalent to the product written in standard form.
I entered y1 5 3x3 1 16x2 1 12x 2 16 and y
2 5 (x 1 2)(3x 2 2)(4 1 x) in my
graphing calculator. Each equation produced the same graph, therefore the
expressions are equivalent.
b. Will multiplying three linear factors always result in a cubic expression?
Explain your reasoning.
Yes. A linear factor has an x-term. The product of three first degree terms is a
third degree term.
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Guiding Questions for Share Phase, Worked Example and Question 2
Is there another way to
multiply binomials?
If the graph of the equation
written in standard form is
not the same as the graph
of the equation written in
factored form, what can
you conclude?
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Grouping
Have students complete
Questions 3 through 5 with a
partner. Then have students
share their responses as
a class.
Guiding Questions for Share Phase, Questions 3 through 5
Which factors did you
multiply together !rst? Does
it matter?
What method did you use to
multiply the binomials?
What effect does a negative
leading term have on the
graph of the cubic function?
What effect does the
constant term have on the
graph of the cubic function?
Does the function pass
through the origin?
If the function passes
through the origin, does this
give you any information
about its factor(s)?
Does the product of a
monomial and two binomials
create a cubic equation?
If the graphs of two or more
functions are different, what
can you conclude?
If the graphs of two or more
functions are the same, do
the functions always have
the same factors?
3. Determine each product. Show all your work and then use a graphing calculator to
verify your product is correct.
a. (x 1 2)(23x 1 2)(1 1 2x)
(x 1 2)(23x 2 6x2 1 2 1 4x)
(x 1 2)(26x2 1 x 1 2)
(26x3 1 x2 1 2x 2 12x2 1 2x 1 4)
26x3 2 11x2 1 4x 1 4
b. (10 1 2x)(5x 1 7)(3x)
(50x 1 70 1 10x2 1 14x)(3x)
(10x2 1 64x 1 70)(3x)
30x3 1 192x 1 210x
310 Chapter 4 Polynomial Functions
4
4. Determine the product of the linear and quadratic factors. Then verify graphically that
the expressions are equivalent.
a. (x 2 6)(2x2 2 3x 1 1)
(2x3 2 3x2 1 x 2 12x2 1 18x 2 6)
2x3 2 15x2 1 19x 2 6
b. (x)(x 1 2)2
x(x2 1 4x 1 4)
x3 1 4x2 1 4x
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4.1 Exploring Cubic Functions 311
5. Max determined the product of three linear factors.
Max
The function f(x) 5 (x 1 2)3 is equivalent to f(x) 5 x3 1 8
a. Explain why Max is incorrect.
The product of (x 1 2)(x 1 2)(x 1 2) is x3 1 6x2 1 12x 1 8.
The functions (x 1 2)3 and x3 1 8 produce different graphs which proves that they
are not equivalent.
b. How many x-intercepts does the function f(x) 5 (x 1 2)3 have? How many zeros?
Explain your reasoning.
The function has only one x-intercept, (22, 0) since it crosses the x-axis only once.
The function has three zeros. The zero x 5 22 is a multiple root, occurring
3 times.