exploring cubic functions - weeblyhhspreapalgebra2.weebly.com/uploads/1/7/0/2/... · 295b chapter 4...

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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

4

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4.1 Exploring Cubic Functions 295

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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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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4

Grouping





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.

600005_A2_TX_Ch04_293-402.indd 298 14/03/14 2:43 PM

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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.


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?


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?


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?


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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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?


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.


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


Questions 6 through 8 with a



as a class.


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?


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




share their responses as

a class.


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.

600005_A2_TX_Ch04_293-402.indd 307 14/03/14 2:43 PM


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?


Yes. A linear factor has an x-term. The product of three first degree terms is a

third degree term.

600005_A2_TX_Ch04_293-402.indd 308 14/03/14 2:43 PM

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


Questions 3 through 5 with a


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


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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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?


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.

exploring cubic functions - weeblyhhspreapalgebra2.weebly.com/uploads/1/7/0/2/... · 295b chapter 4...

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