Functions and Their Graphs

SECTION
3.3

Graphing Techniques: Transformations

SKILLS OBJECTIVES


■��	Sketch the graph of a function using horizontal and vertical shifting of common functions.

■��	Sketch the graph of a function by reflecting a common function about the or .

■��	Sketch the graph of a function by stretching or compressing a common function.

■��	Sketch the graph of a function using a sequence of transformations.

CONCEPTUAL OBJECTIVES


■��	Identify the common functions by their graphs.

■��	Apply multiple transformations of common functions to obtain graphs of functions.

■��	Understand that domain and range also are transformed.

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Horizontal and Vertical Shifts

The focus of the previous section was to learn the graphs that correspond to particular functions such as identity, square, cube, square root, cube root, absolute value, and reciprocal. Therefore, at this point, you should be able to recognize and generate the graphs of

, and

. In this section, we will discuss how to sketch the graphs of functions that are very simple modifications of these functions. For instance, a common function may be shifted (horizontally or vertically), reflected, or stretched (or compressed). Collectively, these techniques are called transformations.

Let's take the absolute value function as an example. The graph of

was given in the last section. Now look at two examples that are much like this function:

and

. Graphing these functions by point-plotting yields

Instead of point-plotting the function

, we could have started with the function

and shifted the entire graph up 2 units. Similarly, we could have generated the graph of the function

by shifting the function

to the right 1 unit. In both cases, the base or starting function is

. Why did we go up for

and to the right for

Note that we could rewrite the functions

and

in terms of

In the case of

, the shift

occurs “outside” the function—that is, outside the parentheses showing the argument. Therefore, the output for

is two more than the typical output for

. Because the output corresponds to the vertical axis, this results in a shift upward of two units. In general, shifts that occur outside the function correspond to a vertical shift corresponding to the sign of the shift. For instance, had the function been

, this graph would have started with the graph of the function

and shifted down two units.

In the case of

, the shift occurs “inside” the function—that is, inside the parentheses showing the argument. Note that the point

that lies on the graph of

was shifted to the point

on the graph of the function

. The

remained the same, but the

shifted to the right one unit. Similarly, the points

and

were shifted to the points

and

, respectively. In general, shifts that occur inside the function correspond to a horizontal shift opposite the sign. In this case, the graph of the function

shifted the graph of the function

to the right one unit. If, instead, we had the function

, this graph would have started with the graph of the function

and shifted to the left one unit.

VERTICAL SHIFTS

Assuming that

is a positive constant,

To Graph	Shift the Graph of
	units upward
	units downward

Adding or subtracting a constant outside the function corresponds to a vertical shift that goes with the sign.

HORIZONTAL SHIFTS

Assuming that

is a positive constant,

To Graph	Shift the Graph of
	units to the left
	units to the right

Adding or subtracting a constant inside the function corresponds to a horizontal shift that goes opposite the sign.

EXAMPLE�1�

Horizontal and Vertical Shifts

Sketch the graphs of the given functions using horizontal and vertical shifts:

(a)��

(b)��

Solution

In both cases, the function to start with is

(a)��

can be rewritten as


1.��	The shift (one unit) occurs outside of the function. Therefore, we expect a vertical shift that goes with the sign.

2.��	Since the sign is negative, this corresponds to a downward shift.

3.��	Shifting the graph of the function down one unit yields the graph of .

(b)��

can be rewritten as


1.��	The shift (one unit) occurs inside of the function. Therefore, we expect a horizontal shift that goes opposite the sign.

2.��	Since the sign is positive, this corresponds to a shift to the left.

3.��	Shifting the graph of the function to the left one unit yields the graph of .

■	YOUR TURN�

Sketch the graphs of the given functions using horizontal and vertical shifts.

(a)��

It is important to note that the domain and range of the resulting function can be thought of as also being shifted. Shifts in the domain correspond to horizontal shifts, and shifts in the range correspond to vertical shifts.

�EXAMPLE�2�

Horizontal and Vertical Shifts and Changes in the Domain and Range

Graph the functions using translations and state the domain and range of each function.

(a)��

(b)��

Solution

In both cases the function to start with is

(a)��

can be rewritten as


1.��	The shift (one unit) is inside the function, which corresponds to a horizontal shift opposite the sign.

2.��	Shifting the graph of to the left one unit yields the graph of . Notice that the point , which lies on the graph of , gets shifted to the point on the graph of .

Although the original function

had an implicit restriction on the domain:

, the function

has the implicit restriction that

. We see that the output or range of

is the same as the output of the original function

(b)��

can be rewritten as


1.��	The shift (two units) is outside the function, which corresponds to a vertical shift with the sign.

2.��	The graph of is found by shifting down two units. Note that the point , which lies on the graph of , gets shifted to the point on the graph of .

The original function

has an implicit restriction on the domain:

. The function

also has the implicit restriction that

. The output or range of

is always two units less than the output of the original function

■	YOUR TURN�

Sketch the graph of the functions using shifts and state the domain and range.

(a)��

(b)��

The previous examples have involved graphing functions by shifting a known function either in the horizontal or vertical direction. Let us now look at combinations of horizontal and vertical shifts.

�EXAMPLE�3�

Combining Horizontal and Vertical Shifts

Sketch the graph of the function

. State the domain and range of

Solution

The base function is


1.��	The shift (one unit) is inside the function, so it represents a horizontal shift opposite the sign.

2.��	The shift is outside the function, which represents a vertical shift with the sign.

3.��	Therefore, we shift the graph of to the left one unit and down two units. For instance, the point on the graph of shifts to the point on the graph of .

■	YOUR TURN�

Sketch the graph of the function

. State the domain and range of

All of the previous transformation examples involve starting with a common function and shifting the function in either the horizontal or the vertical direction (or a combination of both). Now, let's investigate reflections of functions about the

Reflection about the Axes

To sketch the graphs of

and

start by first listing points that are on each of the graphs and then connecting the points with smooth curves.

Note that if the graph of

is reflected about the

, the result is the graph of

. Also note that the function

can be written as the negative of the function

; that is,

. In general, reflection about the

is produced by multiplying a function by

Let's now investigate reflection about the

. To sketch the graphs of

and

start by listing points that are on each of the graphs and then connecting the points with smooth curves.

Note that if the graph of

is reflected about the

, the result is the graph of

. Also note that the function

can be written as

. In general, reflection about the

is produced by replacing

with

in the function. Notice that the domain of

, whereas the domain of

REFLECTION ABOUT THE AXES

The graph of

is obtained by reflecting the graph of

about the

The graph of

is obtained by reflecting the graph of

about the

EXAMPLE�4�

Sketching the Graph of a Function Using Both Shifts and Reflections

Sketch the graph of the function

Solution

Start with the square root function.

Shift the graph of

to the left one unit to arrive at the graph of

Reflect the graph of

about the

to arrive at the graph of

�

�EXAMPLE�5�

Sketching the Graph of a Function Using Both Shifts and Reflections

Sketch the graph of the function

Solution

Start with the square root function.

Shift the graph of

to the left two units to arrive at the graph of

Reflect the graph of

about the

to arrive at the graph of

Shift the graph

up one unit to arrive at the graph of

�

■	YOUR TURN�

Use shifts and reflections to sketch the graph of the function

. State the domain and range of

Look back at the order in which transformations were performed in Example 5: horizontal shift, reflection, and then vertical shift. Let us consider an alternate order of transformations.

Words

Math

Start with the square root function.

Shift the graph of

up one unit to arrive at the graph of

Reflect the graph of

about the

to arrive at the graph of

Replace

with

, which corresponds to a shift of the graph of

to the right two units to arrive at the graph of

In the last step we replaced

with

, which required us to think ahead knowing the desired result was

inside the radical. To avoid any possible confusion, follow this order of transformations:


1.��	Horizontal shifts:

2.��	Reflection: and/or

3.��	Vertical shifts:

Stretching and Compressing

Horizontal shifts, vertical shifts, and reflections change only the position of the graph in the Cartesian plane, leaving the basic shape of the graph unchanged. These transformations (shifts and reflections) are called rigid transformations because they alter only the position. Nonrigid transformations, on the other hand, distort the shape of the original graph. We now consider stretching and compressing of graphs in both the vertical and the horizontal direction.

A vertical stretch or compression of a graph occurs when the function is multiplied by a positive constant. For example, the graphs of the functions

, and

are illustrated below. Depending on if the constant is larger than 1 or smaller than 1 will determine whether it corresponds to a stretch (expansion) or compression (contraction) in the vertical direction.

Note that when the function

is multiplied by 2, so that

, the result is a graph stretched in the vertical direction. When the function

is multiplied by

, so that

, the result is a graph that is compressed in the vertical direction.

VERTICAL STRETCHING AND VERTICAL COMPRESSING OF GRAPHS

The graph of

is found by:


■��	Vertically stretching the graph of


■��	Vertically compressing the graph of

Note:

is any positive real number.

EXAMPLE�6�

Vertically Stretching and Compressing Graphs

Graph the function

Solution

1. Start with the cube function.

2. Vertical compression is expected because

is less than 1.

3. Determine a few points that lie on the graph of

Conversely, if the argument

of a function

is multiplied by a positive real number

, then the result is a horizontal stretch of the graph of

. If

, then the result is a horizontal compression of the graph of

HORIZONTAL STRETCHING AND HORIZONTAL COMPRESSING OF GRAPHS

The graph of

is found by:


■��	Horizontally stretching the graph of


■��	Horizontally compressing the graph of

Note:

is any positive real number.

�EXAMPLE�7�

Vertically Stretching and Horizontally Compressing Graphs

Given the graph of

, graph:

(a)��

(b)��

Solution

(a)��

Since the function is multiplied (on the outside) by 2, the result is that each

is multiplied by 2, which corresponds to vertical stretching.

(b)��

Since the argument of the function is multiplied (on the inside) by 2, the result is that each

is divided by 2, which corresponds to horizontal compression.

■	YOUR TURN�

Graph the function

�EXAMPLE�8�

Sketching the Graph of a Function Using Multiple Transformations

Sketch the graph of the function

Solution

Start with the square function.

Shift the graph of

to the right three units to arrive at the graph of

Vertically stretch the graph of

by a factor of 2 to arrive at the graph of

Reflect the graph

about the

to arrive at the graph of

In Example 8 we followed the same “inside out” approach with the functions to determine the order for the transformations: horizontal shift, vertical stretch, and reflection.

�

SECTION
3.3

SUMMARY

Transformation	To graph the function…	Draw the graph of and then…	Description
Horizontal shifts		Shift the graph of to the left units. Shift the graph of to the right units.	Replace by . Replace by .
Vertical shifts		Shift the graph of up units. Shift the graph of down units.	Add to . Subtract from .
Reflection about the		Reflect the graph of about the .	Multiply by .
Reflection about the		Reflect the graph of about the .	Replace by .
Vertical stretch	, where	Vertically stretch the graph of .	Multiply by .
Vertical compression	, where	Vertically compress the graph of .	Multiply by .
Horizontal stretch	, where	Horizontally stretch the graph of .	Replace by .
Horizontal compression	, where	Horizontally compress the graph of .	Replace by .

SECTION
3.3

EXERCISES

SKILLS

In Exercises 1-12, match the function to the graph.

(a)��

(b)��

(c)��

(d)��

(e)��

(f)��

(g)��

(h)��

(i)��

(j)��

(k)��

(l)��

1.��	l l Shift the graph of up 1 unit.

2.��

3.��	a a Shift the graph of right 1 unit, then reflect over .

4.��

5.��	b b Shift the graph of left 1 unit, then reflect over .

6.��

7.��	i i Shift the graph of right 1 unit, then shift up 1 unit.

8.��

9.��	c c Shift the graph of right 1 unit, then reflect over , and then shift down 1 unit.

10.��

11.��

12.��

In Exercises 13-18, write the function whose graph is the graph of

, but is transformed accordingly.

13.��

Shifted up three units

14.��

Shifted to the left four units

15.��

Reflected about the

16.��

Reflected about the

17.��

Vertically stretched by a factor of 3

18.��

Vertically compressed by a factor of 3

In Exercises 19-24, write the function whose graph is the graph of

, but is transformed accordingly.

19.��

Shifted down four units

20.��

Shifted to the right three units

21.��

Shifted up three units and to the left one unit

22.��

Reflected about the

23.��

Reflected about the

24.��

Reflected about both the

and the

In Exercises 25-48, use the given graph to sketch the graph of the indicated functions.

25.��


(a)��

(b)��

26.��


(a)��

(b)��

27.��


(a)��

(b)��

28.��


(a)��

(b)��

29.��


(a)��

(b)��

30.��


(a)��

(b)��

31.��


(a)��

(b)��

32.��


(a)��

(b)��

33.��

34.��

35.��

36.��

37.��

38.��

39.��

40.��

41.��

42.��

43.��

44.��

45.��

46.��

47.��

48.��

In Exercises 49-74, graph the function using transformations.

49.��

50.��

51.��

52.��

53.��

54.��

55.��

56.��

57.��

58.��

59.��

60.��

61.��

62.��

63.��

64.��

65.��

66.��

67.��

68.��

69.��

70.��

71.��

72.��

73.��

74.��

In Exercises 75-80, transform the function into the form

, where

, and

are constants, by completing the square. Use graph-shifting techniques to graph the function.

75.��

76.��

77.��

78.��

79.��

80.��

APPLICATIONS

81.��

Salary.

A manager hires an employee at a rate of

per hour. Write the function that describes the current salary of the employee as a function of the number of hours worked per week,

. After a year, the manager decides to award the employee a raise equivalent to paying him for an additional 5 hours per week. Write a function that describes the salary of the employee after the raise.

82.��

Profit.

The profit associated with St. Augustine sod in Florida is typically

, where

is the number of pallets sold per year in a normal year. In rainy years Sod King gives away 10 free pallets per year. Write the function that describes the profit of

pallets of sod in rainy years.

83.��

Taxes.

Every year in the United States each working American typically pays in taxes a percentage of his or her earnings (minus the standard deduction). Karen's 2011 taxes were calculated based on the formula

. That year the standard deduction was

and her tax bracket paid

in taxes. Write the function that will determine her 2012 taxes, assuming she receives the raise that places her in the

bracket.

84.��

Medication.

The amount of medication that an infant requires is typically a function of the baby's weight. The number of milliliters of an antiseizure medication

is given by

, where

is the weight of the infant in ounces. In emergencies there is often not enough time to weigh the infant, so nurses have to estimate the baby's weight. What is the function that represents the actual amount of medication the infant is given if his weight is overestimated by 3 ounces?

For Exercises 85 and 86, refer to the following:

Body Surface Area (BSA) is used in physiology and medicine for many clinical purposes. BSA can be modeled by the function

where

is weight in kilograms and

is height in centimeters. Since BSA depends on weight and height, it is often thought of as a function of both weight and height. However, for an individual adult height is generally considered constant; thus BSA can be thought of as a function of weight alone.

85.��

Health/Medicine.

(a) If an adult female is 162 centimeters tall, find her BSA as a function of weight. (b) If she loses 3 kilograms, find a function that represents her new BSA.

86.��

Health/Medicine.

(a) If an adult male is 180 centimeters tall, find his BSA as a function of weight. (b) If he gains 5 kilograms, find a function that represents his new BSA.

CATCH THE MISTAKE

In Exercises 87-90, explain the mistake that is made.

87.��

Describe a procedure for graphing the function

Solution:


(a)��	Start with the function .

(b)��	Shift the function to the left three units.

(c)��	Shift the function up two units.

This is incorrect. What mistake was made?

88.��

Describe a procedure for graphing the function

Solution:


(a)��	Start with the function .

(b)��	Shift the function to the left two units.

(c)��	Reflect the function about the .

(d)��	Shift the function down three units.

This is incorrect. What mistake was made?

89.��

Describe a procedure for graphing the function

Solution:


(a)��	Start with the function .

(b)��	Reflect the function about the .

(c)��	Shift the function to the left three units.

(d)��	Shift the function up one unit.

This is incorrect. What mistake was made?

90.��

Describe a procedure for graphing the function

Solution:


(a)��	Start with the function .

(b)��	Reflect the function about the .

(c)��	Shift the function up one unit.

(d)��	Expand in the vertical direction by a factor of 2.

This is incorrect. What mistake was made?