CorrectionKey=TX-B LESSON 6 . 1 DO NOT EDIT- … · domain and range of cubic functions are the set of all real numbers. The inverse of a cubic function may or may not be a function

Resource LockerA2.2.A ...analyze key attributes such as domain, range, intercepts, symmetries, asymptotic

behavior, and maximum and minimum given an interval. Also A2.6.A.

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Explore 1 Graphing and Analyzing f(x) = x 3

You know that a quadratic function has the standard form ƒ (x) = a x 2 + bx + c where a, b, and c are real numbers and a ≠ 0. Similarly, a cubic function has the standard form ƒ (x) = a x 3 + b x 2 + cx + d where a, b, c and d are all real numbers and a ≠ 0. You can use the basic cubic function, ƒ (x) = x 3, as the parent function for a family of cubic functions related through transformations of the graph of ƒ (x) = x 3.

Complete the table, graph the ordered pairs, and then draw a smooth curve through the plotted points to obtain the graph of ƒ (x) = x 3.

Use the graph to analyze the function and complete the table.

Attributes of f (x) = x 3

Domain ℝ

Range

End behavior As x → + ∞, f (x) → .

As x → -∞, f (x) → .

Zeros of the function x = 0

Where the function has positive values x > 0

Where the function has negative values

Where the function is increasing

Where the function is decreasing The function never decreases.

Is the function even (f (-x) = f (x) ) , odd (f (-x) = -f (x) ) , or neither? , because (x) 3 = .

x y = x 3

-2

-1

0

1

2

0

1

-1

8

-8

Odd

x < 0

- x 3

+∞

ℝ

-∞

The function increases throughout its domain.

Module 6 315 Lesson 1

6 . 1 Graphing Cubic FunctionsEssential Question: How are the graphs of f (x) = a (x - h)

3 + k and f (x) = ( 1 _

b (x - h) )

3

+ k related to the graph of f (x) = x 3 ?

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A2_MTXESE353930_U3M06L1.indd 315 15-01-11 1:52 AM

Texas Math StandardsThe student is expected to:

A2.2.A

Graph the functions f (x ) = √ ― x , f (x ) = 1 __ x , f (x ) = x 3 , f (x) = 3 √― x ,

f (x ) = bx, f (x ) = ⎜x⎟ , and f (x ) = log b (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. Also A2.6.A

Mathematical Processes

A2.1.D

Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Language Objective

2.C.4, 3.B.3, 3.C.4, 3.H.3

Explain to a partner how to predict transformations of a basic cubic function.

HARDCOVER PAGES 225234

Turn to these pages to find this lesson in the hardcover student edition.

Graphing Cubic Functions

ENGAGE Essential Question: How are the graphs of f (x) = a (x - h)

3 + k and

f (x) = ( 1 __ b

(x - h) ) 3 + k related to the

graph of f (x) = x 3 ?Possible answer: Both graphs involve

transformations of the graph of f(x) = x 3 . The first

involves vertically stretching or compressing the

graph of f(x) = x 3 , reflecting it across the x-axis if

a < 0, and translating it horizontally and vertically.

The second involves horizontally stretching or

compressing the graph of f(x) = x 3 , reflecting it

across the y-axis if b < 0, and translating it

horizontally and vertically.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo and how the volume of a spherical aquarium can be described by a cubic function. Then preview the Lesson Performance Task.

315

HARDCOVER PAGES

Turn to these pages to find this lesson in the hardcover student edition.

Resource

Locker

A2.2.A ...analyze key attributes such as domain, range, intercepts, symmetries, asymptotic

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Explore 1 Graphing and Analyzing f(x) = x 3

You know that a quadratic function has the standard form ƒ (x) = a x 2 + bx + c where a, b, and c are real numbers

and a ≠ 0. Similarly, a cubic function has the standard form ƒ (x) = a x 3 + b x 2 + cx + d where a, b, c and d are all

real numbers and a ≠ 0. You can use the basic cubic function, ƒ (x) = x 3, as the parent function for a family of cubic

functions related through transformations of the graph of ƒ (x) = x 3.

Complete the table, graph the ordered pairs,

and then draw a smooth curve through the

plotted points to obtain the graph of ƒ (x) = x 3.

Use the graph to analyze the function and complete the table.

Attributes of f (x) = x 3

Domain

ℝ

Range

End behavior

As x → + ∞, f (x) → .

As x → -∞, f (x) → .

Zeros of the function

x = 0

Where the function has positive values x > 0

Where the function has negative values

Where the function is increasing

Where the function is decreasingThe function never decreases.

Is the function even (f (-x) = f (x) ) , odd

(f (-x) = -f (x) ) , or neither?

, because (x) 3 = .

x y = x 3

-2

-1

0

1

2

0

1

-1

8

-8

Odd

x < 0

- x 3

+∞ℝ

-∞

The function increases

throughout its domain.

Module 6

315

Lesson 1

6 . 1 Graphing Cubic Functions

Essential Question: How are the graphs of f (x) = a (x - h) 3 + k and f (x) = ( 1 _

b (x - h) )

3

+ k

related to the graph of f (x) = x 3 ?

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315 Lesson 6 . 1

L E S S O N 6 . 1

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Reflect

1. How would you characterize the rate of change of the function on the intervals ⎡ ⎣ -1, 0 ⎤ ⎦ and ⎡ ⎣ 0, 1 ⎤ ⎦ compared with the rate of change on the intervals ⎡ ⎣ -2, -1 ⎤ ⎦ and ⎡ ⎣ 1, 2 ⎤ ⎦ ? Explain.

2. A graph is said to be symmetric about the origin (and the origin is called the graph’s point of symmetry) if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Is the graph of ƒ (x) = x 3 symmetric about the origin? Explain.

Explore 2 Predicting Transformations of the Graph of f (x) = x 3

You can apply familiar transformations to cubic functions. Check your predictions using a graphing calculator.

A Predict the effect of the parameter h on the graph of g (x) = (x - h) 3 for each function.

a. The graph of g (x) = (x - 2) 3 is a of the graph of ƒ (x)

[right/up/left/down] 2 units.

b. The graph of g (x) = (x + 2) 3 is a of the graph of ƒ (x)


B Predict the effect of the parameter k on the graph of g (x) = x 3 + k for each function.

a. The graph of g (x) = x 3 + 2 is a of the graph of ƒ (x)


b. The graph of g (x) = x 3 - 2 is a of the graph of ƒ (x)


C Predict the effect of the parameter a on the graph of g (x) = a x 3 for each function.

a. The graph of g (x) = 2 x 3 is a of the graph of ƒ (x) by a

factor of .

b. The graph of g (x) = - 1 __ 2 x 3 is a of the graph of ƒ (x) by a

factor of as well as a across

the .

The rate of change on all four intervals is positive because the function is increasing.

However, the rate of change on the intervals ⎡ ⎣ -1, 0 ⎤ ⎦ and ⎡ ⎣ 0, 1 ⎤ ⎦ is much less than the rate

of change on the intervals ⎡ ⎣ -2, -1 ⎤ ⎦ and ⎡ ⎣ 1, 2 ⎤ ⎦ , because the graph rises more slowly on the

intervals ⎡ ⎣ -1, 0 ⎤ ⎦ and ⎡ ⎣ 0, 1 ⎤ ⎦ than it does on the intervals ⎡ ⎣ -2, -1 ⎤ ⎦ and ⎡ ⎣ 1, 2 ⎤ ⎦ .

Yes, because for every point (x, y) = (x, x 3 ) on the graph of the function, the point

(-x, (-x) 3 ) = (-x, -x 3 ) = (-x, -y) is also on the graph.

translation

translation

translation

translation

1 _ 2

vertical stretch

vertical compression

2

x-axis

reflection


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A2_MTXESE353930_U3M06L1 316 2/22/14 4:45 AM

Math BackgroundA cubic function is a polynomial function of degree 3. The standard form of a cubic function is f (x) = a x 3 + b x 2 + cx + d where a, b, c, and d are real numbers and a ≠ 0. Each function of the family of cubic functions is increasing or decreasing over its entire domain, depending on the sign of the leading coefficient of x 3 . There also may be turning points and relative maximum and relative minimum points, depending on the transformations of the graph of f (x) = x 3 . The domain and range of cubic functions are the set of all real numbers. The inverse of a cubic function may or may not be a function. If the graph has turning points, its inverse is not a function.

EXPLORE 1 Graphing and Analyzing f (x) = x 3

INTEGRATE TECHNOLOGYStudents can use a graphing calculator to graph and generate table values for the

function y = x 3 .

QUESTIONING STRATEGIESHow do you verify that the graph of f(x) = x 3 has symmetry about the

origin? Show that if (x, y) is a point on the graph,

then (-x, -y) is also a point on the graph.

How is the graph of f(x) = x 3 similar to the graph of f(x) = x? The domain and range

for both functions are (-∞, ∞) ; the end behavior

for both functions is f (x) → -∞ as x → -∞, and

f (x) → ∞ as x → +∞.

EXPLORE 2 Predicting Transformations of the Graph of f (x) = x 3

AVOID COMMON ERRORSStudents often assume that the graph of f(x - h) is translated to the left due to the subtraction. For example, if the function is f(x) = (x - 2 ) 3 , students may want to translate f(x) = x 3 two units to the left. Point out that the value of h in f(x - h) is positive, so the transformation is in the positive direction, to the right. Have students graph the two functions to verify this.

CONNECT VOCABULARY Relate the term parent function to parents and children. The children come from the parents; likewise, cubic functions can all be traced back to a parent function.

PROFESSIONAL DEVELOPMENT

Graphing Cubic Functions 316


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D Predict the effect of the parameter b on the graph of g (x) = ( 1 __ b x) 3 for the following values of b.

a. The graph of g (x) = ( 1 __ 2 x) 3 is a of the graph of ƒ (x) by a factor

of .

b. The graph of g (x) = (2x) 3 is a of the graph of ƒ (x) by a

factor of .

c. The graph of g (x) = (- 1 __ 2 x) 3 is a of the graph of ƒ (x) by

a factor of as well as a across

the .

d. The graph of g (x) = (-2x) 3 is a of the graph of ƒ (x) by a

factor of as well as a across

the .

Reflect

3. The graph of g (x) = (-x) 3 is a reflection of the graph of ƒ (x) = x 3 across the y-axis, while the graph of h (x) = - x 3 is a reflection of the graph of ƒ (x) = x 3 across the x-axis. If you graph g(x) and h(x) on a graphing calculator, what do you notice? Explain why this happens.

Explain 1 Graphing Combined Transformations of f (x) = x 3 When graphing transformations of ƒ (x) = x 3 , it helps to consider the effect of the transformations on the three reference points on the graph of ƒ (x) : (-1, -1) , (0, 0) , and (1,1) . The table lists the three points and the corresponding points on the graph of g (x) = a ( 1 __ b (x - h) ) 3 + k. Notice that the point (0, 0) , which is the point of symmetry for the graph of ƒ (x) , is affected only by the parameters h and k. The other two reference points are affected by all four parameters.

f (x) = x 3 g (x) = a ( 1 _ b

(x - h) ) 3

+ k

x y x y

-1 -1 -b + h -a + k

0 0 h k

1 1 b + h a + k

The graphs of g (x) and h (x) are the same because f (x) is an odd function (that is,

f (-x) = -f (x) ) . Given that g (x) = f (-x) and h (x) = -f (x) , you have g (x) = h (x) .

horizontal stretch

2

1 __ 2

1 __ 2

horizontal compression

horizontal compression

2 reflection

y-axis

horizontal stretch

reflection

y-axis



A2_MTXESE353930_U3M06L1.indd 317 2/21/14 3:30 AM

COLLABORATIVE LEARNING

Whole Class ActivityHave groups of students create posters to describe the transformations to the graph of f (x) = x 3 when graphing f (x) = a (x - h) 3 + k. Have them write an example function in the top cell of a graphic organizer. Then, in each cell below the example function, have them list the variable and describe the nature of the transformation.

QUESTIONING STRATEGIESHow is the graph of g(x) = (x - h ) 3 + k related to the parent graph f(x) = x 3 ? The

graph of g (x) = (x - h) 3 + k is the graph of f (x) = x 3

translated h units horizontally and k units vertically.

If a > 0, how is the graph of g(x) = a x 3 related to the graph of f(x) = x 3 ? If a > 1, the graph

of g(x) is the graph of f (x) = x 3 stretched a units

vertically. If 0 < a < 1, the graph of g(x) is the graph

of f(x) = x 3 compressed 1 ____ | a |

units vertically.

EXPLAIN 1 Graphing Combined Transformations of f (x) = x 3

INTEGRATE MATHEMATICAL PROCESSESFocus on PatternsPoint out that to graph the function f (x) = a (x - h) 3 + k, you identify the point of symmetry (h, k) and use the value of a to draw the graph through two additional points (-1 + h, –a + k) and (1 + h, a + k). So, to graph f (x) = 2 (x - 3) 3 + 1, first identify the point of symmetry, (3, 1). Then identify points (-1 + 3, -2 + 1) = (2, -1) , and (1 + 3, 2 + 1) = (4, 3) as additional reference points. A smooth curve through these three points is a good beginning for the graph.

317 Lesson 6 . 1


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Example 1 Identify the transformations of the graph of ƒ (x) = x 3 that produce the graph of the given function g (x) . Then graph g (x) on the same coordinate plane as the graph of ƒ (x) by applying the transformations to the reference points (-1, -1) , (0, 0) , and (1, 1) .

g (x) = 2 (x - 1) 3 - 1

The transformations of the graph of ƒ (x) that produce the graph of g (x) are:

• a vertical stretch by a factor of 2• a translation of 1 unit to the right and 1 unit down

Note that the translation of 1 unit to the right affects only the x-coordinates of points on the graph of ƒ (x) , while the vertical stretch by a factor of 2 and the translation of 1 unit down affect only the y-coordinates.

f (x) = x 3 g (x) = 2 (x - 1) 3 - 1

x y x y

-1 -1 -1 + 1 = 0 2 (-1) - 1 = -3

0 0 0 + 1 = 1 2 (0) - 1 = -1

1 1 1 + 1 = 2 2 (1) - 1 = 1

g (x) = (2 (x + 3) ) 3 + 4

The transformations of the graph of ƒ (x) that produce the graph of g (x) are:

• a horizontal compression by a factor of 1 __ 2 • a translation of 3 units to the left and 4 units up

Note that the horizontal compression by a factor of 1 __ 2 and the translation of 3 units to the left affect only the x-coordinates of points on the graph of ƒ (x) , while the translation of 4 units up affects only the y-coordinates.

f (x) = x 3 g (x) = (2 (x + 3) ) 3 + 4

x y x y

-1 -1 (-1) + = -1 + =

0 0 (0) + = 0 + =

1 1 (1) + = 1 + =

1 __ 2

1 __ 2

1 __ 2

-3 4

4

4 5

4

3

-3 -3

-3

-3 1 __ 2

-2 1 __ 2


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DIFFERENTIATE INSTRUCTION

Critical ThinkingHave students form pairs. Give each pair of students a cubic function in general form and a graphing calculator. Have them graph the parent function f (x) = x 3 and then discuss how their given cubic function will be transformed based the graph of the parent function. Include choices such as reflection across the x-axis, reflection across the y-axis, horizontal and vertical shifts, and horizontal and vertical stretches. Have them graph their given functions to verify their predictions.

QUESTIONING STRATEGIESWhat is the point of symmetry for a graph of the form f(x) = ( 1 __ b (x - h) ) 3 + k? (h, k)

How does a negative value for b affect the graph of f(x) = ( 1 __ b (x - h) ) 3 + k? The graph is

reflected across the y-axis.



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(3, 4)

(2, 1)

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Your Turn

Identify the transformations of the graph of ƒ (x) = x 3 that produce the graph of the given function g (x) . Then graph g (x) on the same coordinate plane as the graph of ƒ (x)by applying the transformations to the reference points (-1, -1) , (0, 0) , and (1, 1) .

4. g (x) = - 1 _ 2 (x - 3) 3

Explain 2 Writing Equations for Combined Transformations of f (x) = x 3

Given the graph of the transformed function g (x) = a ( 1 __ b (x - h) ) 3 + k, you can determine the values of the parameters by using the same reference points that you used to graph g (x) in the previous example.

Example 2 A general equation for a cubic function g (x) is given along with the function’s graph. Write a specific equation by identifying the values of the parameters from the reference points shown on the graph.

g (x) = a (x - h) 3 + k

Identify the values of h and k from the point of symmetry.

(h, k) = (2, 1) , so h = 2 and k = 1.

Identify the value of a from either of the other two reference points.

The rightmost reference point has general coordinates (h + 1, a + k) . Substituting 2 for h and 1 for k and setting the general coordinates equal to the actual coordinates gives this result:

(h + 1, a + k) = (3, a + 1) = (3, 4) , so a = 3.

Write the function using the values of the parameters: g (x) = 3 (x - 2) 3 + 1

The transformations of the graph of f (x) that produce the graph of g (x) are:

• a vertical compression by a factor of 1 __ 2

• a reflection across the x-axis

• a translation of 3 units to the right

Note that the translation of 3 units to the right affects

only the x-coordinates of points on the graph of f (x) ,

while the vertical compression by a factor of 1 _ 2

and the reflection across the x-axis affect

only the y-coordinates. Points: (2, 1 _ 2

), (3, 0), and (4, - 1 _ 2

)




LANGUAGE SUPPORT

Connect VocabularyHave students work in pairs. Instruct one student to explain the effect of any transformation of the parent function f(x) = x 3 . While the first student explains, the second student writes down the steps in the explanation. They switch roles and repeat the procedure for another change, such as the effect of b on the graph of g (x) = 1 __ b x 3 . The goal is to support students as they learn to articulate their understanding—at first, using their own words, then moving towards more mathematically precise explanations.

EXPLAIN 2 Writing Equations for Combined Transformations of f (x) = x 3

AVOID COMMON ERRORSFor students having difficulty identifying the

parameters in the graph of g (x) = a ( 1 __ b (x - h) ) 3 + k,

have them start with identifying the images of the reference points (-1, -1 ) , (0, 0) , (1, 1) from the graph of f(x) = x 3 . Then have them solve for the parameters a, b, h, and k using the image points and (-b + h, -a + k) , (h, k), or (b + h, a + k) , respectively.

319 Lesson 6 . 1


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2

-4

-2

-6

2

x(-1, 1)

(-2, -2)

(-3, -5)

(5, 0)

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B g (x) = ( 1 __ b x - h) 3 + k

Identify the values of h and k from the point of symmetry.

(h, k) = (-4, ) , so h = -4 and k = .

Identify the value of b from either of the other two reference points.

The rightmost reference point has general coordinates

(b + h, 1 + k) . Substituting -4 for h and for k and setting the general coordinates equal to the actual coordinates gives this result:

( b + h, 1 + ) = (b - 4, ) = (-3.5, 2) , so b = .

Write the function using the values of the parameters, and then simplify.

g (x) = ( 1 _____ (x - ) ) 3 +

or

g (x) = ( (x + ) ) 3

+

Your Turn

A general equation for a cubic function g (x) is given along with the function’s graph. Write a specific equation by identifying the values of the parameters from the reference points shown on the graph.

5. g (x) = a (x - h) 3 + k 6. g (x) = ( 1 _ b

(x - h) ) 3 + k

1

(h, k) = (-2, -2) , so h = -2 and k = -2.

(h + 1, a + k) = (-2 + 1, a - 2) = (-1, 1) , so a = 3.

g (x) = 3 (x + 2) 3 - 2

(h, k) = (1, -1) , so h = 1 and k = -1.

(b + h, 1 + k) = (b + 1, 1 - 1) = (5, 0) , so b = 4.

g (x) = ( 1 _ 4 (x - 1) ) 3

- 1

1 1

1 2 0.5

0.5-4 1

2 4 1





QUESTIONING STRATEGIESWhy does the general form f (x) = a (x - h) 3 + k or

f (x) = ( 1 __ b (x - h) ) 3 + k need to be specified before finding the equation of the transformed function? The general form must be specified so

that a unique equation can be found.



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ℓ

Length (cm)

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Explain 3 Modeling with a Transformation of f (x) = x 3 You may be able to model a real-world situation that involves volume with a cubic function. Sometimes mass may also be involved in the problem. Mass and volume are related through density, which is defined as an object’s mass per unit volume. If an object has mass m and volume V, then its density d is d = m __ V . You can rewrite the formula as m = dV to express mass in terms of density and volume.

Example 3 Use a cubic function to model the situation, and graph the function using calculated values of the function. Then use the graph to obtain the indicated estimate.

Estimate the length of an edge of a child’s alphabet block (a cube) that has a mass of 23 g and is made from oak with a density of 0.72 g/c m 3 .

Let ℓ represent the length (in centimeters) of an edge of the block. Since the block is a cube, the volume V (in cubic centimeters) is V (ℓ) = ℓ 3 . The mass m (in grams) of the block is m (ℓ) = 0.72 · V (ℓ) = 0.72 ℓ 3 . Make a table of values for this function.

Length (cm) Mass (g)

0 0

1 0.72

2 5.76

3 19.44

4 46.08

Draw the graph of the mass function, recognizing that the graph is a vertical compression of the graph of the parent cubic function by a factor of 0.72. Then draw the horizontal line m = 23 and estimate the value of ℓ where the graphs intersect.

The graphs intersect where ℓ ≈ 3.2, so the edge length of the child’s block is about 3.2 cm.



A2_MTXESE353930_U3M06L1 321 2/22/14 4:46 AM

EXPLAIN 3 Modeling with a Transformation of f (x ) = x 3

QUESTIONING STRATEGIESHow is the density formula for a sphere of radius r, m (r) = DV (r) with mass m, density

D, and volume V related to the volume of a sphere, V(r) = 4 __ 3 π r 3 , with radius r? The mass of the sphere

is the product of the density and the volume, so

m (r) = 4 __ 3 πD r 3 .

What kind of transformation does the equation m (r) = 4 __ 3 πD r 3 show? a vertical

stretch of the function m (r) = r 3 by a factor of 4 __ 3 πD

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B Estimate the radius of a steel ball bearing with a mass of 75 grams and a density of 7.82 g/c m 3 .

Let r represent the radius (in centimeters) of the ball bearing. The volume V (in cubic centimeters) of the ball bearing is

V (r) = r 3 . The mass m (in grams) of the ball

bearing is m (r) = 7.82 · V (r) = r 3 .

Radius (cm) Mass (g)

0

0.5

1

1.5

2

Draw the graph of the mass function, recognizing that the graph is a vertical

of the graph of the parent cubic function by a factor of . Then draw the

horizontal line m = and estimate the value of r where the graphs intersect.

The graphs intersect where r ≈ , so the radius of the steel ball bearing is about

cm.

Reflect

7. Discussion Why is it important to plot multiple points on the graph of the volume function.

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Because you are using the graph to estimate the value of the independent variable from

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Use a cubic function to model the situation, and graph the function using calculated values of the function. Then use the graph to obtain the indicated estimate.

8. Polystyrene beads fill a cube-shaped box with an effective density of 0.00076 kg/c m 3 (which accounts for the space between the beads). The filled box weighs 6 kilograms while the empty box had weighed 1.5 kilograms. Estimate the inner edge length of the box.

Elaborate

9. Identify which transformations (stretches or compressions, reflections, and translations) of ƒ (x) = x 3 change the following attributes of the function.

a. End behavior

b. Location of the point of symmetry

c. Symmetry about a point

10. Essential Question Check-In Describe the transformations you must perform on the graph of ƒ (x) = x 3 to obtain the graph of g (x) = a (x - h) 3 + k.

Let ℓ represent the length (in centimeters) of an inner edge of the box. Since the box is a cube, the volume V (in cubic centimeters) is V (ℓ) = ℓ 3 . The mass m (in kilograms) of the polystyrene beads in the box is m (ℓ) = 0.00076 · V (ℓ) = 0.00076 ℓ 3 . Make a table of values for this function.

Length (cm) Mass (kg)

0 0

5 0.095

10 0.76

15 2.565

20 6.08

Draw the graph of the mass function, recognizing that the graph is a vertical compression of the graph of the parent cubic function by a factor of 0.00076. Then draw the horizontal line m = 6 - 1.5 = 4.5 and estimate the value of ℓ where the graphs intersect.

The graphs intersect where ℓ ≈ 18, so the inner edge length of the box is about 18 cm.

Reflections across the x-axis (a < 0) and reflections across the y-axis (b < 0) change

the end behavior.

Vertical translations (k ≠ 0) and horizontal translations (h ≠ 0) change the location of

the point of symmetry.

No transformations change the function’s symmetry about a point.

If a < 0, reflect the parent graph across the x-axis. Then either stretch the graph vertically

by a factor of a if ⎜a⎟ > 1 or compress the graph vertically by a factor of a if ⎜a⎟ < 1. Finally,

translate the graph h units horizontally and k units vertically.


DO NOT EDIT--Changes must be made through "File info" CorrectionKey=TX-B

A2_MTXESE353930_U3M06L1.indd 323 1/12/15 9:39 PM

ELABORATE INTEGRATE MATHEMATICAL PROCESSESFocus on Critical ThinkingBefore students graph any cubic function, encourage them to predict how the graph will look based on the leading coefficient of x when the function is written in general form g (x) = a ( 1 __ b (x - h) ) 3 + k, and on the values of h and k.

SUMMARIZE THE LESSONHow do you use reference points to graph or write equations for combined transformations

of f (x) = x 3 ? Use the reference points (-1, -1),

(0, 0), and (1, 1). Under a combined transformation

(-1, -1) becomes (-b + h, -a + k), (0, 0) becomes

(h, k), and (1, 1) becomes (b + h, a + k). To graph the

combined transformation, you apply the

transformations to the reference points and use

those transformed points to graph the transformed

function. To write an equation, you use the

transformations of the reference points to solve for

a, b, h, and k and use those values to write the

equation of the transformed function.

323 Lesson 6 . 1


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• Online Homework• Hints and Help• Extra Practice

1. Graph the parent cubic function ƒ (x) = x 3 and use the graph to answer each question.

a. State the function’s domain and range.

b. Identify the function’s end behavior.

c. Identify the graph’s x- and y-intercepts.

d. Identify the intervals where the function has positive values and where it has negative values.

e. Identify the intervals where the function is increasing and where it is decreasing.

f. Tell whether the function is even, odd, or neither. Explain.

g. Describe the graph’s symmetry.

Describe how the graph of g(x) is related to the graph of ƒ (x) = x 3 .

2. g (x) = (x - 4) 3 3. g (x) = -5 x 3

4. g (x) = x 3 + 2 5. g (x) = (3x) 3

Evaluate: Homework and Practice

The domain is ℝ, and the range is ℝ.

As x → +∞, f (x) → +∞. As x → -∞, f (x) → -∞.

The graph’s only x-intercept is 0. The graph’s only y-intercept is 0.

The function has positive values on the interval (0, +∞) and negative values on the interval (-∞, 0) .

The function is odd because f (-x) = (-x) 3 = - x 3 = -f (x) .

The graph is symmetric about the origin.

The function is increasing throughout its domain. The function never decreases.

translation of the graph of f (x) right 4 units.

vertical stretch of the graph of f (x) by a factor of 5 and a reflection across the x-axis.

translation of the graph of f (x) up 2 units.

horizontal compression of the

graph of f (x) by a factor of 1 _ 3 .




A2_MTXESE353930_U3M06L1.indd 324 2/21/14 3:30 AMExercise Depth of Knowledge (D.O.K.) Mathematical Processes

1–10 1 Recall of Information 1.F Analyze relationships

11–21 2 Skills/Concepts 1.F Analyze relationships

22–24 3 Strategic Thinking 1.A Everyday life

25 2 Skills/Concepts 1.F Analyze relationships

26 3 Strategic Thinking 1.G Explain and justify arguments

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore 1Graphing and Analyzing f (x) = x 3

Exercise 1

Explore 2Predicting Transformations of the Graph of f (x) = x 3

Exercises 2–9

Example 1Graphing Combined Transformations of f (x) = x 3

Exercises 10–17

Example 2Writing Equations for Combined Transformations of f (x) = x 3

Exercises 18–21

Example 3Modeling with a Transformation off (x) = x 3

Exercises 22–23

INTEGRATE TECHNOLOGYWhen graphing a cubic function or any other function on a graphing calculator, students

should choose a good viewing window: one that displays the important characteristics of the graph. As a class, have students brainstorm what should show on the graphing screen. Sample answer: the end

behavior; the turning point, or point of symmetry of

the graph; the x-axis of the graph



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6. g (x) = (x + 1) 3 7. g (x) = 1 _ 4 x 3

8. g (x) = x 3 - 3 9. g (x) = (- 2 _ 3 x) 3

Identify the transformations of the graph of ƒ (x) = x 3 that produce the graph of the given function g(x). Then graph g(x) on the same coordinate plane as the graph of ƒ (x) by applying the transformations to the reference points (-1, -1) , (0, 0) , and (1, 1) .

10. g (x) = ( 1 _ 3 x) 3 11. g (x) = 1 _ 3 x 3

12. g (x) = (x - 4) 3 - 3 13. g (x) = (x + 1) 3 + 2

translation of the graph of f (x) left 1 unit.

vertical compression of the graph of

f (x) by a factor of 1 _ 4 .

translation of the graph of f (x) down 3 units.

horizontal stretch of the graph of f (x) by

a factor of 3 _ 2 as well as a reflection across

the y-axis.

a horizontal stretch by a factor of 3. Reference points on the graph of g(x) are (-3, -1), (0, 0), and (3, 1).

a vertical compression by a factor of 1 _ 3 .

Reference points on the graph of g(x) are

(-1, - 1 _ 3 ) , (0, 0) and (1, 1 _ 3 ) .

translation of 4 units to the right and 3 units down. Reference points on the graph of g(x) are (3, -4) , (4, -3) , and (5, -2) .

a translation of 1 unit to the left and 2 units up. Reference points on the graph of g(x) are (-2, 1) , (-1, 2) , and (0, 3) .




AVOID COMMON ERRORSStudents may be confused about where to start when finding the equation for a combined transformation of the graph of f(x) = x 3 . Explain that they should compare the corresponding points of the graph to the reference points of the graph of f(x) = x 3 . If the graph is stretched or compressed horizontally, then they may use the form f (x) = ( 1 __ b (x - h) ) 3 + k. If the graph is stretched or compressed vertically, then they may use the form f (x) = a (x - h) 3 + k.

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14. g (x) = -2 x 3 + 3 15. g (x) = (-2 (x - 2) ) 3

16. g (x) = -3 (x + 1) 3 - 1 17. g (x) = (- 1 _ 4 (x - 1) ) 3 + 2

•averticalstretchbyafactorof2

•areflectionacrossthex-axis

•atranslationof3unitsup

Referencepointsonthegraphofg(x)are(-1,5),(0,3),and(1,1).

•ahorizontalcompressionbyafactorof1_2•areflectionacrossthey-axis

•atranslationof2unitstotheright

Referencepointsonthegraphofg(x)are

(11_2,1),(2,0),and(21_2,-1).

•averticalstretchbyafactorof3

•areflectionacrossthex-axis

• atranslationof1unittotheleftand1unitdown

Referencepointsonthegraphofg(x)are(-2,2),(-1,-1),and(0,-4).

•ahorizontalstretchbyafactorof4

•areflectionacrossthey–axis

• atranslationof1unittotherightand2unitsup

Referencepointsonthegraphofg(x)are(-3,3),(1,2),and(5,1).





INTEGRATE MATHEMATICAL PROCESSESFocus on ModelingSuggest that students make a graphic organizer similar to the one below to help them remember the differences in the types of transformations of f(x) = x 3 . Suggest students choose one or more cubic functions from this exercise set and give an example of each type of transformation.

Transformation Example

Vertical shift

Horizontal shift

Vertical stretch

Horizontal compression



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A general equation for a cubic function g (x) is given along with the function’s graph. Write a specific equation by identifying the values of the parameters from the reference points shown on the graph.

18. g (x) = ( 1 _ b

(x - h) ) 3 + k 19. g (x) = a (x - h) 3 + k

20. g (x) = ( 1 _ b

(x - h) ) 3 + k 21. g (x) = a (x - h) 3 + k

(h, k) = (-2, -3) , so h = -2 and k = -3.

(b + h, 1 + k) = (b - 2, 1 - 3) = (1, -2) , so b = 3.

g (x) = ( 1 _ 3 (x + 2) ) 3

- 3

(h, k) = (2, -1) , so h = 2 and k = -1.

(b + h, 1 + k) = (b + 2, 1 - 1) = (1.5, 0) ,

so b = -0.5.

g (x) = ( 1 ____ -0.5 (x - 2) ) 3

- 1 = (-2 (x - 2) ) 3 - 1

(h, k) = (1, 4) , so h = 1 and k = 4.

(h + 1, a + k) = (1 + 1, a + 4) = (2, 1) , so a = -3.

g (x) = -3 (x - 1) 3 + 4

(h, k) = (-1, 2) , so h = -1 and k = 2.

(h + 1, a + k) = (-1 + 1, a + 2) = (0, 2.5) ,

so a = 0.5.

g (x) = 0.5 (x + 1) 3 + 2




INTEGRATE MATHEMATICAL PROCESSESFocus on Math ConnectionsPoint out that another name for the point of symmetry of a cubic function in this lesson is the inflection point. This is the point for which the “curvature” of the graph changes. For a cubic equation with a positive leading coefficient, the rate of change in the y-values of the graph decreases as x approaches this point from below and then begins to increase again as x increases past the point. In later mathematics classes, students will find that the slope of the curve at this point is 0.

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Use a cubic function to model the situation, and graph the function using calculated values of the function. Then use the graph to obtain the indicated estimate.

22. Estimate the edge length of a cube of gold with a mass of 1 kg. The density of gold is 0.019 kg/ cm 3 .

23. A proposed design for a habitable Mars colony is a hemispherical biodome used to maintain a breathable atmosphere for the colonists. Estimate the radius of the biodome if it is required to contain 5.5 billion cubic feet of air.

The volume V (in cubic centimeters) of the cube is V (ℓ) = ℓ 3 . The mass m (in kilograms) of the cube is m (ℓ) = 0.019 · V (ℓ) = 0.019 ℓ 3 .

Length (cm) Mass (kg)

0 0

1 0.019

2 0.152

3 0.513

4 1.216

The graphs intersect where ℓ ≈ 3.75, so the edge length of the cube of gold is about 3.75 centimeters.

Since the biodome is a hemisphere, the volume V (in billions of cubic feet) of

the biodome is V (r) = 1 _ 2 ( 4 _ 3 π r 3 ) ≈ 2.09 r 3 .

Radius (thousands of feet)

Volume (billions of cubic feet)

0 0

0.5 0.26

1 2.09

1.5 7.05

2 16.72

The graphs intersect where r ≈ 1.4, so the radius of the biodome is about 1.4 thousand (1400) ft.





INTEGRATE MATHEMATICAL PROCESSESFocus on ModelingFor modeling problems involving geometry, review the formulas for the volumes of a sphere and rectangular prism. Students should recall that the volume V of a sphere can be found with the formula V = 4 __ 3 π r 3 , where r is the radius, and the volume of a rectangular prism is V = lwh, where l is the length of the rectangular base, w is the width of the base, and h is the height of the prism.



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24. Multiple Response Select the transformations of the graph of the parent cubic function that result in the graph of g (x) = (3 (x - 2) ) 3 + 1.

A. Horizontal stretch by a factor of 3 E. Translation 1 unit up

B. Horizontal compression by a factor of 1 _ 3 F. Translation 1 unit down

C. Vertical stretch by a factor of 3 G. Translation 2 units left

D. Vertical compression by a factor of 1 _ 3 H. Translation 2 units right

H.O.T. Focus on Higher Order Thinking

25. Justify Reasoning Explain how horizontally stretching (or compressing) the graph of ƒ (x) = x 3 by a factor of b can be equivalent to vertically compressing (or stretching) the graph of ƒ (x) = x 3 by a factor of a.

26. Critique Reasoning A student reasoned that g (x) = (x - h) 3 can be rewritten as g (x) = x 3 - h 3 , so a horizontal translation of h units is equivalent to a vertical translation of - h 3 units. Is the student correct? Explain.

Horizontally stretching or compressing the graph of the parent cubic function by a factor

of b results in the graph of the function g (x) = ( 1 _ b

x) 3

. You can rewrite g(x) as follows:

g (x) = ( 1 _ b

x) 3

= ( 1 _ b

) 3

· x 3

= 1 ___ b 3

· x 3

The function is now in the form g (x) = a x 3 where a = 1 __ b 3

. Notice that when ⎜b⎟ > 1, ⎜a⎟ < 1,

which means that a horizontal stretch is equivalent to a vertical compression. Similarly,

when ⎜b⎟ < 1, ⎜a⎟ > 1, which means that a horizontal compression is equivalent to a

vertical stretch.

No, the student isn’t correct. There is no property of exponents that allows you to distribute an exponent to a sum or difference, so g (x) = (x - h)

3 cannot be rewritten in the form g (x) = x 3 + k. From a

geometric perspective, it doesn’t make sense to think that there can be a vertical translation of a graph that is equivalent to a horizontal translation of the graph except in the trivial case where the graph is not moved at all (h = k = 0) .




CONNECT VOCABULARY To help students remember how the words shift, stretch, and compression are used in the context of this lesson, remind students that a shift is a translation, a transformation that does not change the shape or size of the original figure as the figure moves on the coordinate plane. A stretch (or compression), on the other hand, represents what happens to the graph of the parent figure if the leading coefficient of x is not equal to one. Reference points from the transformed graph will have more space between the corresponding y-values (“stretched”) than the parent graph, or less space between the y-values (“compressed”) than points on the parent graph. You can see these changes on the graph for small values of x.

PEERTOPEER DISCUSSIONHave students work in pairs. Instruct one student in each pair to sketch the graph of a cubic function of the form f(x) = a (x - h) 3 + k, while the other gives verbal instructions for each step. Then have the student who sketched the graph write the steps that were followed. Have students switch roles and repeat the exercise using a cubic function of the form f(x) = 1 __ b (x - h ) 3 + k.

JOURNALHave students compare and contrast algebraic representations of vertical and horizontal transformations of cubic functions.

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Julio wants to purchase a spherical aquarium and fill it with salt water, which has an average density of 1.027 g/c m 3 . He has found a company that sells four sizes of spherical aquariums.

Aquarium Size Diameter (cm)

Small 15

Medium 30

Large 45

Extra large 60

a. If the stand for Julio’s aquarium will support a maximum of 50 kg, what is the largest size tank that he should buy? Explain your reasoning.

b. Julio’s friend suggests that he could buy a larger tank if he uses fresh water, which has a density of 1.0 g/c m 3 . Do you agree with the friend? Why or why not?

Lesson Performance Task

a. The volume V (in cubic centimeters) of a sphere with diameter d (in centimeters) is

V (d) = 4 _ 3 π ( d _ 2 ) 3

= π __ 6 d 3 . If the sphere is filled with salt water having a density of 1.027 g/cm3,

then the mass m (in kilograms) of the salt water is m (V) = 1.027 ____ 1000 V = 0.001027V. The table

summarizes the volume and mass of salt water for each size of tank.

Aquarium Size Volume (c m 3 ) Mass of Water (kg)

Small 1,760 1.81

Medium 14,100 14.5

Large 47,700 49.0

Extra large 113,000 116

Some students may say that Julio should buy the large tank, because the mass of the water,

49.0 kg, is less than 50 kg. Other students may say that he should buy the medium tank,

because he has to consider the mass of the empty tank, which will likely make the total mass

greater than 50 kg.

b. Students’ answers will depend on how they answered Part a. If they answered that Julio

should buy the large tank, then the minor difference in density will have no effect on his

choice. If they said he should buy the medium tank, some students may decide that the

lesser density of fresh water might reduce the mass of the water just enough to allow him to

buy the large tank.




EXTENSION ACTIVITY

Julio decides he wants to fill one-third of his aquarium with sand that has a density of 1.6 g/c m 3 . The remaining two-thirds of the aquarium will be filled with water. Have students answer the questions in the Performance Task using this new information. Then have them graph the mass in the aquarium compared with the aquarium size, both with and without the sand. Ask them how the addition of the sand affects the total mass in the aquarium and if this information will help Julio decide between a Medium and a Large tank.

AVOID COMMON ERRORSMake sure students are multiplying the volume by the density instead of dividing by the density. Have students write the units next to the values as they do the calculation. If the units cancel properly, the final units should be mass. Ask students what the real-life consequence of dividing by the density would be.

QUESTIONING STRATEGIESWhy would using fresh water allow Julio to buy a larger tank? Fresh water has a lower

density and therefore has a lower mass for the same

volume.

If the diameter of the Medium tank is twice the diameter of the Small tank, why isn’t the

volume of the Medium tank twice that of the Small tank? The volume is described by a cubic function.

The ratio of the volumes is the ratio of their

diameters cubed.

Why is it important to know the density of the water? The density is the mass per volume. If

the volume is known, then the density will give the

mass that the volume will hold.

Scoring Rubric

2 points: The student’s answer is an accurate and complete execution of the task or tasks.

1 point: The student’s answer contains attributes of an appropriate response but is flawed.

0 points: The student’s answer contains no attributes of an appropriate response.



Documents

CorrectionKey=TX-B LESSON 6 . 1 DO NOT EDIT- … · domain and range of cubic functions are the set of all real numbers. The inverse of a cubic function may or may not be a function