section 1.3: basic graphs and · pdf file... for example, we have the input-output model and...

(Section 1.3: Basic Graphs and Symmetry) 1.3.1

SECTION 1.3: BASIC GRAPHS and SYMMETRY

LEARNING OBJECTIVES

• Know how to graph basic functions.

• Organize categories of basic graphs and recognize common properties,

such as symmetry.

• Identify which basic functions are even / odd / neither and relate this to

symmetry in their graphs.

PART A: DISCUSSION

• We will need to know the basic functions and graphs in this section without

resorting to point-plotting.

• To help us remember them, we will organize them into categories. What are the

similarities and differences within and between categories, particularly with

respect to shape and symmetry in graphs? (We will revisit symmetry in Section 1.4

and especially in Section 1.7.)

• A power function f has a rule of the form f x( ) = xn , where the exponent or

power n is a real number.

• We will consider graphs of all power functions with integer powers, and

some power functions with non-integer powers.

• In the next few sections, we will manipulate and combine these building blocks to

form a wide variety of functions and graphs.


PART B: CONSTANT FUNCTIONS

If f x( ) = c , where c is a real number, then f is a constant function.

• Any real input yields the same output, c.

If f x( ) = 3, for example, we have the input-output model and the flat graph of

y = 3, a horizontal line, below.

PART C: IDENTITY FUNCTIONS

If f x( ) = x , then f is an identity function.

• Its output is identical to its input.

6 f 6

10 f 10

• There are technically different identity functions on different domains.

The graph of y = x is the line below.


PART D: LINEAR FUNCTIONS

If f x( ) = mx + b , where m and b are real numbers, and m 0 ,

then f is a linear function.

In Section 0.14, we graphed y = mx + b as a line with slope m and y-intercept b.

If f x( ) = 2x 1, for example, we graph the line with slope 2 and y-intercept 1.

PART E: SQUARING FUNCTION and EVEN FUNCTIONS

Let f x( ) = x2 . We will construct a table and graph f .

x

f x( ) Point

x

f x( ) Point

0 0 0, 0( ) 0 0

0, 0( )

1 1 1,1( )

1 1

1,1( )

2 4 2, 4( )

2 4

2, 4( )

3 9 3, 9( )

3 9

3, 9( )


TIP 1: The graph never falls below the x-axis, because squares of real

numbers are never negative.

Look at the table. Each pair of opposite x values yields a common function

value f x( ) , or y.

• Graphically, this means that every point x, y( ) on the graph has a

“mirror image partner”

x, y( ) that is also on the graph. These

“mirror image pairs” are symmetric about the y-axis.

• We say that f is an even function. (Why?)

A function f is even f x( ) = f x( ) , x Dom f( )

The graph of y = f x( ) is

symmetric about the y -axis.

Example 1 (Even Function: Proof)

Let f x( ) = x2 . Prove that f is an even function.

§ Solution

Dom f( ) = . x ,

f x( ) = x( )2

= x2

= f x( )

Q.E.D. (Latin: Quod Erat Demonstrandum)

• This signifies the end of a proof. It means “that which was to

have been proven, shown, or demonstrated.”

TIP 2: Think: If we replace x with

x( ) as the input, we obtain

equivalent outputs. §


PART F: POWER FUNCTIONS WITH POSITIVE, EVEN POWERS and

INTERSECTION POINTS

The term “even function” comes from the following fact:

If f x( ) = xn , where n is an even integer, then f is an even function.

• The graph of y = x2 is called a parabola (see Chapters 2 and 10).

• The graphs of y = x4 , y = x6 , etc. resemble that parabola, although

they are not called parabolas.

• We will discuss the cases with nonpositive exponents later.

How do these graphs compare?

For example, let f x( ) = x2 and

g x( ) = x4 . Compare the graphs of f and g.

Their relationship when x > 1 is unsurprising:

x

f x( )

x2

g x( )

x4

2

4 16

3

9 81

4

16 256

• As expected, x4> x2

if x > 1. As a result, the graph of y = x4 lies

above the graph of y = x2 on the x-interval 1,( ) .


However, their relationship on the x-interval

0,1 might be surprising:

x

f x( )

x2

g x( )

x4

0

0 0

0.1

0.01 0.0001

1

3

1

9

1

81

1

2

1

4

1

16

1

1 1

• WARNING 1: As it turns out, x4< x2

on the x-interval 0,1( ) .

As a result, the graph of y = x4 lies below the graph of y = x2 on that

x-interval.

• Since the graphs have the points 0, 0( ) and 1,1( ) in common, those

points are intersection points.

Graphically, here’s what we have (so far) on the x-interval

0, ) .

Below, f x( ) = x2 and

g x( ) = x4 .


How can we quickly get the other half of the picture? Exploit symmetry!

f and g are both even functions, so their graphs are symmetric about

the y-axis.

• Observe that

1,1( ) is our third intersection point.

• In calculus, you might find the area of one or both of those tiny regions

bounded (trapped) by the graphs.

Let h x( ) = x6 . How does the graph of h below compare?

• The graph of h rises even faster than the others as we move far away

from x = 0 , but it is even flatter than the others close to x = 0 .


PART G: POWER FUNCTIONS WITH POSITIVE, ODD POWERS and

ODD FUNCTIONS

Let f x( ) = x3 . We will construct a table and graph f .

x

f x( ) Point

x

f x( ) Point

0 0 0, 0( ) 0 0

0, 0( )

1 1 1,1( )

1

1

1, 1( )

2 8 2, 8( )

2

8

2, 8( )

3 27 3, 27( )

3

27

3, 27( )


Look at the table. Each pair of opposite x values yields opposite function

values. That is, f x( ) and f x( ) are always opposites.

• Graphically, this means that every point x, y( ) on the graph has a

“mirror image partner”

x, y( ) on the other side of the origin.

The two points are separated by a 180 rotation (a half revolution)

about the origin. These “mirror image pairs” are symmetric about

the origin.

• We say that f is an odd function. (Why?)

A function f is odd f x( ) = f x( ) , x Dom f( )

The graph of y = f x( ) is

symmetric about the origin.

• In other words, if the graph of f is rotated 180 about the origin,

we obtain the same graph.

Example 2 (Odd Function: Proof)

Let f x( ) = x3 . Prove that f is an odd function.

§ Solution

Dom f( ) = . x ,

f x( ) = x( )3

= x3

= x3( )= f x( )

Q.E.D.

TIP 3: Think: If we replace x with

x( ) as the input, we obtain

opposite outputs. §


The term “odd function” comes from the following fact:

If f x( ) = xn , where n is an odd integer, then f is an odd function.

• The graphs of y = x5 , y = x7 , etc. resemble the graph of y = x3 .

• In Part C, we saw that the graph of y = x is a line.

• We will discuss the cases with negative exponents later.

How do these graphs compare?

For example, let f x( ) = x3 and

g x( ) = x5 . Compare the graphs of f and g.

Based on our experience from Part F, we expect that the graph of g

rises or falls even faster than the graph of f as we move far away

from x = 0 , but it is even flatter than the graph of f close to x = 0 .

WARNING 2: Zero functions are functions that only output 0 (Think: f x( ) = 0 ).

Zero functions on domains that are symmetric about 0 on the real number line are

the only functions that are both even and odd. (Can you show this?)

WARNING 3: Many functions are neither even nor odd.


PART H : f x( ) = x

0

Let f x( ) = x0 . What is f 0( ) ? It is agreed that 02= 0 and 2

0= 1 , but what is 0

0?

Different sources handle the expression 00 differently.

• If 00 is undefined, then f x( ) = 1 x 0( ) , and f has the graph below.

•• There is a hole at the point 0,1( ) .

• There are many reasons to define 00 to be 1. For example, when analyzing

polynomials, it is convenient to have x0= 1 for all real x without having to

consider x = 0 as a special case. Also, this will be assumed when we discuss

the Binomial Theorem in Section 9.6.

•• Then, f x( ) = 1 on , and f has the graph below.

• In calculus, 00 is an indeterminate limit form. An expression consisting of a base

approaching 0 raised to an exponent approaching 0 may, itself, approach a real number

(not necessarily 0 or 1) or not. The expression 00 is called indeterminate by some

sources.

In any case, f is an even function.


PART I: RECIPROCAL FUNCTION and

POWER FUNCTIONS WITH NEGATIVE, ODD POWERS

Let f x( ) =

1

xor x 1( ) .

We call f a reciprocal function, because its output is the reciprocal

(or multiplicative inverse) of the input.

We will carefully construct the graph of f .

Let’s construct a table for x 1.

x 1 10 100

f x( ) , or 1

x 1

1

10

1

100 0+

• The 0+

notation indicates an approach to 0 from greater numbers,

without reaching 0.

The table suggests the following graph for x 1:

The x-axis is a horizontal asymptote (“HA”) of the graph.

An asymptote is a line that a curve approaches in a “long-run” or

“explosive” sense. The distance between them approaches 0.

• Asymptotes are often graphed as dashed lines, although some

sources avoid dashing the x- and y-axes.

• Horizontal and vertical asymptotes will be formally defined in Section 2.9.


Let’s now construct a table for 0 < x 1 .

x 0

+

1

100

1

10 1

f x( ) , or 1

x 100 10 1

• We write: 1

x as x 0+

(“1

x approaches infinity as

x approaches 0 from the right, or from greater numbers”).

•• In the previous table,

1

x0+ as x . Graphically,

1

x

approaches 0 “from above,” though we say “from the right.”

• We will revisit this notation and terminology when we discuss limits

in calculus in Section 1.5.

We now have the following graph for x > 0 :

The y-axis is a vertical asymptote (“VA”) of the graph.

How can we quickly get the other half of the picture? Exploit symmetry!

f is an odd function, so its graph is symmetric about the origin.

TIP 4: Reciprocals of negative real numbers are negative real

numbers. 0 has no real reciprocal.


The graph exhibits opposing behaviors about the vertical asymptote (“VA”).

• The function values increase without bound from the right of the VA,

and they decrease without bound from the left of the VA.

The graph of y =1

xor x 1( ) , or xy = 1, is called a hyperbola (see Chapter 10).

The graphs of y =1

x3or x 3( ) , y =

1

x5or x 5( ) , etc. resemble that hyperbola, but

they are not called hyperbolas.

Below, f x( ) =

1

x yields the blue graph;

g x( ) =

1

x3 yields the red graph.

• The graph of g approaches the x-axis more rapidly as x and as x .

• The graph of g approaches the y-axis more slowly as x 0+

and as x 0 (“as x approaches 0 from the left, or from lesser numbers”).

This is actually because the values of g “explode” more rapidly.

• When we investigate the graph of y =

1

x2 in Part J, we will understand these

behaviors better.


PART J: POWER FUNCTIONS WITH NEGATIVE, EVEN POWERS

Let h x( ) =

1

x2or x 2( ) . We will compare the graph of h to the graph of y =

1

x.

Let’s construct a table for x 1.

x 1 10 100

f x( ) , or 1

x 1

1

10

1

100 0

h x( ) , or 1

x2 1

1

100

1

10,000 0

• This suggests that the graph of h approaches the x-axis more rapidly as x .

The table suggests the following graphs for x 1:

The x-axis is a horizontal asymptote (“HA”) of the graph of h.

Let’s now construct a table for 0 < x 1 .

x 0

+

1

100

1

10 1

f x( ) , or

1

x 100 10 1

h x( ) , or 1

x2 10,000 100 1

• This suggests that the graph of h approaches the y-axis more slowly as x 0+ .

This is actually because the values of h “explode” more rapidly.


We now have the following graphs for x > 0 :

The y-axis is a vertical asymptote (“VA”) of the graph of h.

How can we quickly get the other half of the graph of h? Exploit symmetry!

h is an even function, so its graph is symmetric about the y-axis.

TIP 5: This graph lies entirely above the x-axis, because

1

x2 is always

positive in value for nonzero values of x.

The graph exhibits symmetric behaviors about the vertical asymptote (“VA”).

• The function values increase without bound from the left and from the

right of the VA.

The graphs of y = x 4 or 1

x4, y = x 6 or

1

x6, etc. resemble the graph above.


PART K: SQUARE ROOT FUNCTION

Let f x( ) = x or x1/2( ) . We discussed the graph of f in Section 1.2.

WARNING 4: f is not an even function, because it is undefined for x < 0 .

The graphs of y = x

4or x1/4( ) ,

y = x

6or x1/6( ) , etc. resemble this graph, as do

the graphs of y = x34or x3/4( ) , y = x58

or x5/8( ) , etc. (See Footnote 1.)

PART L: CUBE ROOT FUNCTION

Let f x( ) = x

3or x1/3( ) . The graph of f resembles the graph in Part K for x 0 .

WARNING 5: The cube root of a negative real number is a negative real number.

Dom f( ) = .

f is an odd function; its graph is symmetric about the origin.

The graphs of y = x

5or x1/5( ) ,

y = x

7or x1/7( ) , etc. resemble this graph, as do

the graphs of y = x35

or x3/5( ) , y = x59

or x5/9( ) , etc. (See Footnote 2.)


PART M : f x( ) = x

2/3

Let f x( ) = x23or x2/3( ) . The graph of f resembles the graphs in Parts K and L

for x 0 .

f is an even function; its graph below is symmetric about the y-axis.

• WARNING 6: Some graphing utilities omit the part of the graph to the left

of the y-axis.

• In calculus, we will call the point at 0, 0( ) a cusp, because:

•• it is a sharp turning point for the graph, and

•• as we approach the point from either side, we approach

±( ) “infinite steepness.”

The graphs of y = x25

or x2/5( ) , y = x47

or x4/7( ) , etc. resemble the graph above.

(See Footnote 3.)


PART N: ABSOLUTE VALUE FUNCTION

We discussed the absolute value operation in Section 0.4.

The piecewise definition of the absolute value function (on ) is given by:

f x( ) = x =x, if x 0

x, if x < 0

• We will discuss more piecewise-defined functions in Section 1.5.

f is an algebraic function, because we can write: f x( ) = x = x2 .

WARNING 7: Writing x2 as x2/2

would be inappropriate if it is

construed as x, which would not be equivalent for x < 0 , or as

x( )2

,

which has domain

0, ) . (See Footnote 4.)

f is an even function, so its graph will be symmetric about the y-axis.

The graph of y = x for x 0 has a mirror image in the graph of y = x for x 0 .

• In calculus, we will call the point at 0, 0( ) a corner, because:

•• the graph makes a sharp turn there, and

•• the point is not a cusp.

(A corner may or may not be a turning point where the graph changes from rising

to falling, or vice-versa.)


PART O: UPPER SEMICIRCLES

In Section 1.2, we saw that the graph of x2+ y2

= 9 y 0( ) is the upper half of the

circle of radius 3 centered at 0, 0( ) .

• Solving for y, we obtain: y = 9 x2 .

More generally, the graph of x2+ y2

= a2 y 0( ) , where a > 0 , is an upper

semicircle of radius a.

• Solving for y, we obtain: y = a2 x2 .

Let f x( ) = a2 x2 . f is an even function, so its upper semicircular graph

below is symmetric about the y-axis.


PART P: A GALLERY OF GRAPHS

TIP 6: If you know the graphs well, you don’t have to memorize the domains,

ranges, and symmetries. They can be inferred from the graphs.

• In the Domain and Range column,

\ 0{ } denotes the set of nonzero real

numbers. In interval form,

\ 0{ } is

, 0( ) 0,( ) .

Function

Rule

Type of

Function

(Sample)

Graph

Domain;

Range

Even/Odd;

Symmetry

f x( ) = c Constant

;

c{ } Even;

y-axis

f x( ) = x

Identity

(Type of

Linear)

;

Odd;

origin

f x( ) = mx + b

m 0( )

Linear

;

Odd

b = 0 ;

then, origin

f x( ) = x2

xn : n 2, even( )

Power

;

0, )

Even;

y-axis

f x( ) = x3

xn : n 3, odd( )

Power

;

Odd;

origin

f x( ) = x0 Power

See

Part H

See

Part H

Even;

y-axis

f x( ) = x 1 or

1

x

xn : n < 0, odd( )

Power

\ 0{ } ;

\ 0{ }

Odd;

origin

f x( ) = x 2 or1

x2

xn : n < 0, even( )

Power

\ 0{ } ;

0,( )

Even;

y-axis


Function

Rule

Type of

Function

(Sample)

Graph

Domain;

Range

Even/Odd;

Symmetry

f x( ) = x1/2 or x

x

n: n 2, even( )

Power

0, ) ;

0, ) Neither

f x( ) = x1/3 or x

3

x

n: n 3, odd( )

Power

;

Odd;

origin

f x( ) = x2/3 Power

;

0, ) Even;

y-axis

f x( ) = x Absolute

Value

(Algebraic)

;

0, ) Even;

y-axis

f x( ) = a2 x2

a > 0( )

(Type of

Algebraic)

a, a ;

0, a

Even;

y-axis


FOOTNOTES

1. Power functions with rational powers of the form

odd

even.

Let f x( ) = xN / D

, where N is an odd and positive integer, and D is an even and positive integer.

f x( ) = x1/ 2 f x( ) = x3/ 2

• If

N

D is a proper fraction (where N < D ), then the graph of f is concave down and

resembles the graph on the left. Examples:

f x( ) = x or x1/ 2( ) ,

f x( ) = x34

or x3/ 4( ) .

• If

N

D is an improper fraction (where N > D ), then the graph of f is concave up and

resembles the graph on the right. Examples:

f x( ) = x3 or x3/ 2( ) ,

f x( ) = x74

or x7 / 4( ) .


odd

odd.

Let f x( ) = xN / D, where N and D are both odd and positive integers.

f x( ) = x1/3

f x( ) = x3/3

= x f x( ) = x9/3

= x3

• If

N

D is a proper fraction, then the graph of f resembles the leftmost graph.

Examples: f x( ) = x

3or x1/3( ) , f x( ) = x35

or x3/5( ) .

• If

N

D is an improper fraction where N > D , then the graph of f resembles the

rightmost graph. For example, f x( ) = x9/3

= x3.

• If N = D (

N

D is still improper), then we obtain the line y = x (see the middle graph)

as a “borderline” case. For example, f x( ) = x3/3

= x .



even

odd.

Let f x( ) = xN / D

, where N is an even and positive integer, and D is an odd and positive integer.

f x( ) = x2/3

f x( ) = x6/3

= x2

• If

N

D is a proper fraction, then the graph of f resembles the graph on the left.

Examples: f x( ) = x23

or x2/3( ) , f x( ) = x47

or x4/7( ) .

• If

N

D is an improper fraction, then the graph of f resembles the graph on the right,

where f x( ) = x63

= x6/3= x2 .


even

even.

Let f x( ) = xN / D

, where N and D are both even and positive integers.

Different interpretations of xN / D lead to different approaches to Dom f( ) .

• For example, let f x( ) = x2/6

.

•• If x 0 , then f x( ) = x2/6

= x1/3, or x3

.

•• If x2/6 is interpreted as x26

, then x2/6 is real-valued, even if x < 0 .

Under this interpretation, Dom f( ) = .

•• If x2/6

is interpreted as

x6( )

2

, then x2/6

is not real-valued when x < 0 .

Under this interpretation, Dom f( ) = 0, ) .

(Section 1.4: Transformations) 1.4.1

SECTION 1.4: TRANSFORMATIONS

LEARNING OBJECTIVES

• Know how to graph transformations of functions.

• Know how to find an equation for a transformed basic graph.

• Use graphs to determine domains and ranges of transformed functions.

PART A: DISCUSSION

• Variations of the basic functions from Section 1.3 correspond to variations of the

basic graphs. These variations are called transformations.

• Graphical transformations include rigid transformations such as translations

(“shifts”), reflections, and rotations, and nonrigid transformations such as

vertical and horizontal “stretching and squeezing.”

• Sequences of transformations correspond to compositions of functions, which we

will discuss in Section 1.6.

• After this section, we will be able to graph a vast repertoire of functions, and we

will be able to find equations for many transformations of basic graphs.

• We will relate these ideas to the standard form of the equation of a circle with

center h, k( ) , which we saw in Section 0.13. In the Exercises, the reader can revisit

the Slope-Intercept Form of the equation of a line, which we saw in Section 0.14.

• We will use these ideas to graph parabolas in Section 2.2 and conic sections in

general in Chapter 10, as well as trigonometric graphs in Chapter 4.

• Thus far, y and f x( ) have typically been interchangeable. This will no longer be

the case in many of our examples.


PART B: TRANSLATIONS (“SHIFTS”)

Translations (“shifts”) are transformations that move a graph without changing its

shape or orientation.

Let G be the graph of y = f x( ) .

Let c be a positive real number.

Vertical Translations (“Shifts”)

The graph of y = f x( )+ c is G shifted up by c units.

• We are increasing the y-coordinates.

The graph of y = f x( ) c is G shifted down by c units.

Horizontal Translations (“Shifts”)

The graph of y = f x c( ) is G shifted right by c units.

The graph of y = f x + c( ) is G shifted left by c units.

Example 1 (Translations)

Let f x( ) = x . Its graph, G, is the center graph in purple below.


A table can help explain how these translations work.

In the table, “und.” means “undefined.”

x

f x( )

x

f x( ) + 2

x + 2

f x( ) 2

x 2

f x 2( )

x 2

f x + 2( )

x + 2

3 und. und. und. und. und.

2 und. und. und. und. 0

1 und. und. und. und. 1

0 0 2 2 und. 2

1 1 3 1 und. 3

2 2 2 + 2 2 2 0 2

3 3 3 + 2 3 2 1 5

How points

change

y-coords.

increase

2 units

y-coords.

decrease

2 units

x-coords.

increase

2 units

x-coords.

decrease

2 units

G moves …

UP DOWN RIGHT LEFT

§

Example 2 (Finding Domain and Range; Revisiting Example 1)

We can infer domains and ranges of the transformed functions in Example 1

from the graphs and the table in Example 1.

Let f x( ) = x . Then, Dom f( ) = Range f( ) = 0, ) .

Let g x( ) = x + 2 x 2 x 2 x + 2

Dom g( )

Think: x 0, ) 0, ) 2, ) 2, )

Range g( )

Think: y

2, )

2, )

0, )

0, )

§


WARNING 1: Many people confuse the horizontal shifts.

• Compare the x-intercepts of the graphs of y = x and y = x 2 .

The x-intercept is at x = 0 for the first graph, while it is at x = 2 for the

second graph. The fact that the point 0, 0( ) lies on the first graph implies

that the point 2, 0( ) lies on the second graph.

• More generally: The point a, b( ) lies on the first graph the point

a + 2, b( ) lies on the second graph. Therefore, the second graph is obtained

by shifting the first graph to the right by 2 units.

PART C: REFLECTIONS

Reflections

Let G be the graph of y = f x( ) .

The graph of y = f x( ) is G reflected about the x-axis.

The graph of y = f x( ) is G reflected about the y-axis.

The graph of y = f x( ) is G reflected about the origin.

• This corresponds to a 180 rotation (half revolution) about the

origin. It combines both transformations above, in either order.

Example 3 (Reflections)

Again, let f x( ) = x .


A table can help explain how these reflections work.

In the table, “und.” means “undefined.”

x

f x( )

x

f x( )

x

f x( )

x

f x( )

x

3 und. und. 3 3

2 und. und. 2 2

1 und. und. 1

1

0

0

0 0 0

1 1 1 und. und.

2 2 2 und. und.

3 3 3 und. und.

Points are

reflected about x-axis y-axis

Both, or

origin

§

Example 4 (Finding Domain and Range; Revisiting Example 3)

We can infer domains and ranges of the transformed functions in Example 3

from the graphs and the table in Example 3.

Let f x( ) = x . Then,

Dom f( ) = Range f( ) = 0, ) .

Let g x( ) = x x x

Dom g( )

Think: x

0, )

, 0(

, 0(

Range g( )

Think: y

, 0(

0, )

, 0(

WARNING 2: x is defined as a real value for nonpositive real values

of x, because the opposite of a nonpositive real value is a nonnegative real

value. §


Example 5 (Reflections and Symmetry)

Let f x( ) = x2 . The graph of f is below.

The graph is its own reflection about the y-axis, because f is an even

function. The graphs of y = f x( ) and

y = f x( ) are the same:

f x( ) = x( )

2

= x2 . Thus, the graph is symmetric about the y-axis. §

Example 6 (Reflections and Symmetry)

Let f x( ) = x3 . The graph of f is below.

• The graph is its own reflection about the origin, because f is an

odd function. The graphs of y = f x( ) and

y = f x( ) are the same:

f x( ) = x( )3

= x3 . The graph is symmetric about the origin. §


PART D: NONRIGID TRANSFORMATIONS; STRETCHING AND SQUEEZING

Nonrigid transformations can change the shape of a graph beyond a mere

reorientation, perhaps by stretching or squeezing, unlike rigid transformations

such as translations, reflections, and rotations.

If f is a function, and c is a real number, then cf is called a constant multiple of f .

The graph of y = cf x( ) is:

a vertically stretched version of G if c > 1

a vertically squeezed version of G if 0 < c < 1

The graph of y = f cx( ) is:

a horizontally squeezed version of G if c > 1

a horizontally stretched version of G if 0 < c < 1

If c < 0 , then perform the corresponding reflection either before or after the

vertical or horizontal stretching or squeezing.

WARNING 3: Just as for horizontal translations (“shifts”), the cases involving

horizontal stretching and squeezing may be confusing. Think of c as an

“aging factor.”

Example 7 (Vertical Stretching and Squeezing)

Let f x( ) = x . First consider the form

y = cf x( ) .


• For any x-value in

0, ) , such as 1, the corresponding y-coordinate for

the y = x graph is doubled to obtain the y-coordinate for the y = 2 x

graph. This is why there is vertical stretching.

• Similarly, the graph of y =

1

2x exhibits vertical squeezing , because the

y-coordinates have been halved. §

Example 8 (Horizontal Stretching and Squeezing; Revisiting Example 7)

Again, let f x( ) = x . Now consider the form

y = f cx( ) .

The graph of y = f 4x( ) is the graph of y = 2 x in blue, because:

f 4x( ) = 4x = 2 x . The vertical stretching we described in Example 7

may now be interpreted as a horizontal squeezing.

(This is not true of all functions.)

• The function value we got at x = 1 we now get at x =

1

4.

The graph of y = f1

4x is the graph of

y =

1

2x in red, because:

f1

4x =

1

4x =

1

2x . The vertical squeezing we described in Example 7

may now be interpreted as a horizontal stretching.

• The function value we got at x = 1 we now get at x = 4 . §


PART E: SEQUENCES OF TRANSFORMATIONS

Example 9 (Graphing a Transformed Function)

Graph y = 2 x + 3 .

§ Solution

• We may want to rewrite the equation as y = x + 3 + 2 to more clearly indicate the vertical shift.

• We will “build up” the right-hand side step-by-step. Along the way, we

transform the corresponding function and its graph.

• We begin with a basic function with a known graph. (Point-plotting should

be a last resort.) Here, it is a square root function. Let f1

x( ) = x .

Basic Graph: y = x Graph: y = x + 3

Begin with: f1

x( ) = x Transformation: f

2x( ) = f

1x + 3( )

Effect: Shifts graph left by 3 units

Graph: y = x + 3 Graph: y = x + 3 + 2

Transformation: f

3x( ) = f

2x( ) Transformation:

f

4x( ) = f

3x( ) + 2

Effect: Reflects graph about x-axis Effect: Shifts graph up by 2 units


WARNING 4: We are expected to carefully trace the movements of any

“key points” on the developing graphs. Here, we want to at least trace the

movements of the endpoint. We may want to identify intercepts, as well.

Why is the y-intercept of our final graph at 2 3 , or at 0, 2 3( )?

Why is the x-intercept at 1, or at 1, 0( )? (Left as exercises for the reader.) §

Example 10 (Finding an Equation for a Transformed Graph)

Find an equation for the transformed basic graph below.

§ Solution

The graph appears to be a transformation of the graph of the absolute value

function from Section 1.3, Part N.

Basic graph: y = x

Begin with: f1

x( ) = x


There are different strategies that can lead to a correct equation.

Strategy 1 (Raise, then reflect)

Effect: Shifts graph up by 1 unit Effect: Reflects graph about x-axis

Transformation: f

2x( ) = f

1x( ) +1 Transformation:

f x( ) = f

2x( )

Graph: y = x +1 Graph:

y = x +1( )

• WARNING 5: It may help to write f x( ) = f

2x( ) , since it

reminds us to insert grouping symbols.

Possible answers: f x( ) = x +1( ) , or f x( ) = x 1.

Strategy 2 (Reflect, then drop)

Effect: Reflects graph about x-axis Effect: Shifts graph down by 1 unit

Transformation: f

2x( ) = f

1x( ) Transformation:

f x( ) = f

2x( ) 1

Graph: y = x Graph:

y = x 1

Possible answer: f x( ) = x 1, which we saw in Strategy 1.


Strategy 3 (Switches the order in Strategy 2, but this fails!)

Basic graph: y = x

Begin with: f1

x( ) = x

Effect: Shifts graph down by 1 unit Effect: Reflects graph about x-axis

Transformation: f

2x( ) = f

1x( ) 1 Transformation:

f

3x( ) = f

2x( )

Graph: y = x 1 Graph: y = x 1( )

Observe that y = x 1( ) is not equivalent to our previous

answers.

WARNING 6: The order in which transformations are applied can

matter, particularly when we mix different types of transformations. §


PART F: TRANSLATIONS THROUGH COORDINATE SHIFTS

Translations through Coordinate Shifts

A graph G in the xy-plane is shifted h units horizontally

and k units vertically.

• If h < 0 , then G is shifted left by h units.

• If k < 0 , then G is shifted down by k units.

To obtain an equation for the new graph, take an equation for G and:

• Replace all occurrences of x with x h( ) , and

• Replace all occurrences of y with

y k( ) .

Example 11 (Translating a Circle through Coordinate Shifts;

Revisiting Section 0.13)

We want to translate the circle in the xy-plane with radius 3 and center 0, 0( )

so that its new center is at

2,1( ) . Find the standard form of the equation of

the new circle.

§ Solution

We take the equation x2+ y2

= 9 for the old black circle and:

• Replace x with

x 2( )( ) , or x + 2( ) , and

• Replace y with y 1( ) .

This is because we need to shift the black circle left 2 units and up 1 unit to

obtain the new red circle.


Answer: x + 2( )2

+ y 1( )2

= 9 . §

• We will use this technique in Section 2.2 and Chapter 10 on conic sections.

Equivalence of Translation Methods for Functions

Consider the graph of y = f x( ) . A coordinate shift of h units horizontally

and k units vertically yields an equation that is equivalent to one we

would have obtained from our previous approach:

y k = f x h( )y = f x h( ) + k

• Think: h , k , if h and k are positive numbers.

• We will revisit this form when we study parabolas in Section 2.2.

Example 12 (Equivalence of Translation Methods for Functions)

We will shift the first graph to the right by 2 units and up 1 unit.

Graph of y = x2 Graph of y 1= x 2( )

2

, or

y = x 2( )

2

+1

§

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