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Graphing Transformations of Sine and Cosine Functions

Author: Sophia

what's covered

In this lesson, you will graph the sine and cosine functions in the xy-plane. Specifically, this lesson will cover:

1. The Graphs of the Basic Sine and Cosine Functions
2. Transformations of Graphs of Sine and Cosine Functions
3. Writing the Equation of a Sinusoidal Function From Its Graph

1. The Graphs of the Basic Sine and Cosine Functions

Consider the unit circle shown below, with all points corresponding to special angles and multiples of special angles labeled.

As theta increases from one quadrantal angle to the next, the behavior of sin theta is shown in the table.

As Increases From:	(y-coordinate):
0 to	increases from 0 to 1
to	decreases from 1 to 0
to $fraction numerator 3 straight pi over denominator 2 end fraction$	decreases from 0 to -1
$fraction numerator 3 straight pi over denominator 2 end fraction$ to	increases from -1 to 0

We’re going to use this information to sketch the graph of the sine function in the xy-coordinate system.

Let y equals sin x. The table above means that the graph of has the following key points:

open parentheses 0 comma space 0 close parentheses

open parentheses straight pi over 2 comma space 1 close parentheses

open parentheses straight pi comma space 0 close parentheses

$open parentheses fraction numerator 3 straight pi over denominator 2 end fraction comma space short dash 1 close parentheses$ open parentheses 2 straight pi comma space 0 close parentheses

open parentheses 2 straight pi comma space 0 close parentheses

Here is a table of those points along with points from the first quadrant of the unit circle to help us understand the shape of the graph.

	0						$fraction numerator 3 straight pi over denominator 2 end fraction$
	0		$fraction numerator square root of 2 over denominator 2 end fraction$	$fraction numerator square root of 3 over denominator 2 end fraction$	1	0	-1	0

Using these points, the graph of y equals sin x on the interval open square brackets 0 comma space 2 straight pi close square brackets is shown below.

This is the most important period of the sine graph, since all other periods repeat this one. Note again the key points on the graph:

Key Point				$open parentheses fraction numerator 3 straight pi over denominator 2 end fraction comma space short dash 1 close parentheses$
Description	Starting point, x-intercept	Maximum	x-intercept	Minimum point	x-intercept

The following are properties of one period of the basic sine graph:

Halfway between endpoints of one period, an x-intercept occurs.
Halfway between the first two x-intercepts, the maximum value occurs.
Halfway between the second two x-intercepts, a minimum value occurs.

Recall that sin open parentheses x plus-or-minus 2 straight pi close parentheses equals sin x.

Because of this relationship, we say that the sine function has a period of 2 straight pi comma

meaning that the values of sin x

are equal when x is increased or decreased by a multiple of 2 straight pi.

As a result, the complete graph of f open parentheses x close parentheses equals sin x

is shown below.

The graph continues in this pattern indefinitely. Since there are no breaks or holes in the graph, the domain of f open parentheses x close parentheses equals sin x is the set of all real numbers, also written in interval notation as open parentheses short dash infinity comma space infinity close parentheses. The range of this function is open square brackets short dash 1 comma space 1 close square brackets since the smallest output value is -1 and the largest output value is 1.

Recall from an earlier lesson that the sine function is odd, meaning that sin open parentheses short dash x close parentheses equals short dash sin x. In the xy-coordinate system, this means that the graph is symmetric with respect to the origin.

Since y equals 0 is the “middle value,” we call the line the horizontal midline, which is the line that represents the average of the maximum and minimum values of the function.

Now, let’s examine the cosine function.

As theta increases from one quadrantal angle to the next, the behavior of cos theta is shown in the table.

As Increases From:	(x-coordinate):
0 to	decreases from 1 to 0
to	decreases from 0 to -1
to $fraction numerator 3 straight pi over denominator 2 end fraction$	increases from -1 to 0
$fraction numerator 3 straight pi over denominator 2 end fraction$ to	increases from 0 to 1

We’re going to use this information to sketch the graph of the cosine function in the xy-coordinate system.

Let y equals cos x. The table above means that the graph of has the following key points:

open parentheses 0 comma space 1 close parentheses

open parentheses straight pi over 2 comma space 0 close parentheses

open parentheses straight pi comma space short dash 1 close parentheses

$open parentheses fraction numerator 3 straight pi over denominator 2 end fraction comma space 0 close parentheses$ open parentheses 2 straight pi comma space 1 close parentheses

open parentheses 2 straight pi comma space 1 close parentheses

Here is a table of those points, along with points from the first quadrant of the unit circle to help us understand the shape of the graph.

	0						$fraction numerator 3 straight pi over denominator 2 end fraction$
	1	$fraction numerator square root of 3 over denominator 2 end fraction$	$fraction numerator square root of 2 over denominator 2 end fraction$		0	-1	0	1

Recall that cos open parentheses x plus-or-minus 2 straight pi close parentheses equals cos x. Because of this relationship, we say that the cosine function has a period of 2 straight pi comma meaning that the values of cos x are equal when x is increased or decreased by a multiple of 2 straight pi. As a result, the complete graph of y equals cos x is shown below.

Recall from an earlier lesson that the cosine function is even, meaning that cos open parentheses short dash x close parentheses equals cos x. In the xy-coordinate system, this means that the graph is symmetric with respect to the y-axis.

The most important period of the cosine graph is the one that starts at x equals 0 and ends at x equals 2 straight pi comma since all other periods repeat this one. Note again the key points on the graph:

Key Point				$open parentheses fraction numerator 3 straight pi over denominator 2 end fraction comma space 0 close parentheses$
Description	Starting point, maximum value	x-intercept	Minimum point	Ending point, x-intercept	Maximum value

The following are properties of one period of the basic cosine graph:

Halfway between endpoints (maximum values), the minimum value occurs.
Halfway between the first maximum and the minimum value, an x-intercept occurs.
Halfway between the minimum and the second maximum value, another x-intercept occurs.

The graph continues in this pattern indefinitely. Since there are no breaks or holes in the graph, the domain of y equals cos x

is the set of all real numbers, also written in interval notation as open parentheses short dash infinity comma space infinity close parentheses.

open parentheses short dash infinity comma space infinity close parentheses.

The range of this function is open square brackets short dash 1 comma space 1 close square brackets

since the smallest output value is -1 and the largest output value is 1.

The graphs of sine and cosine functions are used to model several real-life situations, such as heights of ocean waves and seasonal temperatures. Graphs that resemble sine and cosine graphs are called sinusoidal graphs, or sinusoids.

big idea

The sine and cosine functions both have domain open parentheses short dash infinity comma space infinity close parentheses

and range

Both functions have a period equal to 2 straight pi.

The horizontal midline of each graph is y equals 0.

terms to know

Period of a Function

The smallest value of p for which f open parentheses x plus p close parentheses equals f open parentheses x close parentheses.

In other words, if the graph of f open parentheses x close parentheses

is shifted horizontally by p units to the left or to the right, the result is the original graph.

Midline

A horizontal line that represents the value that is the average between the maximum value and the minimum value.

Sinusoidal Graphs (Sinusoids)

Any graph that resembles the graph of a sine or cosine function.

2. Transformations of Graphs of Sine and Cosine Functions

A sinusoidal function has equation y equals a sin open parentheses b x minus c close parentheses plus d or y equals a cos open parentheses b x minus c close parentheses plus d comma where a comma b, c, and d are real numbers with a not equal to 0 and b not equal to 0.

In the next two parts, we’ll see what the roles of a comma b, c, and d have in shaping the graphs of y equals a sin open parentheses b x minus c close parentheses plus d or y equals a cos open parentheses b x minus c close parentheses plus d comma as compared to y equals sin x and y equals cos x , respectively.

2a. Vertical Stretches and Compressions, Amplitude

Recall that for some nonzero constant a comma the graph of y equals a times f open parentheses x close parentheses is a vertical stretch of the graph of y equals f open parentheses x close parentheses if open vertical bar a close vertical bar greater than 1 and a vertical compression if If a less than 0 comma then the graph is also reflected over the x-axis.

Consider the graph of f open parentheses x close parentheses equals 2 sin x. Its graph, along with the graph of y equals sin x comma is shown below.

The graph of f open parentheses x close parentheses equals 2 sin x is a vertical stretch of the graph of y equals sin x by a factor of 2. Note that the domain is open parentheses short dash infinity comma space infinity close parentheses comma the range is open square brackets short dash 2 comma space 2 close square brackets comma and the period is 2 straight pi.

Now consider the graph of f open parentheses x close parentheses equals short dash 2 sin x. Its graph, along with the graph of y equals sin x comma is shown below.

The graph of f open parentheses x close parentheses equals short dash 2 sin x is a vertical stretch of the graph of y equals sin x by a factor of 2 and is also reflected over the x-axis. Note that the domain is open parentheses short dash infinity comma space infinity close parentheses comma the range is open square brackets short dash 2 comma space 2 close square brackets comma and the period is 2 straight pi.

Lastly, consider the graph of f open parentheses x close parentheses equals 1 half cos x. Its graph, along with the graph of y equals cos x comma is shown below.

The graph of f open parentheses x close parentheses equals 1 half cos x is a vertical compression of the graph of y equals cos x by a factor of 1 half. Note that the domain is open parentheses short dash infinity comma space infinity close parentheses comma the range is open square brackets short dash 1 half comma space 1 half close square brackets comma and the period is 2 straight pi.

To better describe the vertical stretch factor, we define the amplitude of a sinusoidal graph.

Then, the graphs of y equals sin x and y equals cos x have amplitude 1.

Now consider the graphs of f open parentheses x close parentheses equals a sin x and From the examples above, when a equals 2 or a equals short dash 2 comma the amplitude is 2. When a equals 1 half comma the amplitude is 1 half.

big idea

The graphs of f open parentheses x close parentheses equals a sin x

and

f open parentheses x close parentheses equals a cos x

have amplitude open vertical bar a close vertical bar.

term to know

Amplitude

The maximum vertical distance that the graph is away from its horizontal midline.

2b. Horizontal Stretches and Compressions, Period

We’ll now investigate sinusoidal graphs in the form y equals a sin open parentheses b x close parentheses and y equals a cos open parentheses b x close parentheses comma where b greater than 0.

Consider the function y equals cos open parentheses 2 x close parentheses.

The graphs of y equals cos x and y equals cos open parentheses 2 x close parentheses on the interval open square brackets 0 comma space 2 straight pi close square brackets are shown below.

The graph of y equals cos open parentheses 2 x close parentheses undergoes two complete cycles between x equals 0 and x equals 2 straight pi and its first complete cycle between and x equals straight pi.

This means that the period of y equals cos open parentheses 2 x close parentheses is straight pi.

In general, suppose b greater than 0 colon

Since the angle is bx, one full cycle occurs when
Since divide all three parts by b, which gives $0 less or equal than x less or equal than fraction numerator 2 straight pi over denominator b end fraction.$

formula to know

Period of a Sine or Cosine Function

Assuming

the graph of either y equals sin open parentheses b x close parentheses

y equals cos open parentheses b x close parentheses

has period $fraction numerator 2 straight pi over denominator b end fraction.$

hint

then the even/odd identities can be used to rewrite the trigonometric function so that the angle contains a positive coefficient of the variable.

EXAMPLE

Consider the equation y equals short dash 3 cos open parentheses 1 fourth x close parentheses.

Since the coefficient of the cosine term is negative, the graph is reflected over the x-axis.
Since the amplitude is 3, the minimum value of the function is -3 and the maximum value of the function is 3.
Since the coefficient of x is the period of the function is $fraction numerator 2 straight pi over denominator open parentheses begin display style 1 fourth end style close parentheses end fraction equals 8 straight pi.$

Two periods of the graph of the function are shown below.

Let’s now examine the period from x equals 0

Since the graph is reflected over the x-axis, the period starts and ends at its minimum value rather than its maximum value. These points are and
Since the graph is reflected over the x-axis, the maximum value occurs halfway between and which is at
Halfway between the minimum and maximum values, the x-intercepts occur. These are at the points and

try it

Consider the equation y equals 1 half sin open parentheses 3 x close parentheses.

Identify the period and amplitude.

The coefficient of the sine function is 1 half comma

which is positive. This means the amplitude is also 1 half.

The period of a sinusoidal graph is $fraction numerator 2 straight pi over denominator b end fraction comma$ where b is the coefficient of x. Thus, the period of this function is $fraction numerator 2 straight pi over denominator 3 end fraction.$

Identify the x-intercepts, and the points where the minimum and maximum values occur on the interval corresponding to the period that starts with x = 0.

The period of the function is $fraction numerator 2 straight pi over denominator 3 end fraction.$

Since the “starting point” of a sine graph is open parentheses 0 comma space 0 close parentheses comma

open parentheses 0 comma space 0 close parentheses comma

we know that the end of its period is at the point $open parentheses fraction numerator 2 straight pi over denominator 3 end fraction comma space 0 close parentheses.$ We also know that there is another x-intercept halfway between the endpoints, which is located at open parentheses straight pi over 3 comma space 0 close parentheses.

open parentheses straight pi over 3 comma space 0 close parentheses.

Thus, in increasing order, the x-intercepts are at open parentheses 0 comma space 0 close parentheses comma

open parentheses straight pi over 3 comma space 0 close parentheses comma

and $open parentheses fraction numerator 2 straight pi over denominator 3 end fraction comma space 0 close parentheses.$

The maximum will be halfway between the first two x-intercepts, open parentheses 0 comma space 0 close parentheses

and

or at

This results in a value of y equals 1 half.

The maximum will occur at open parentheses straight pi over 6 comma space 1 half close parentheses.

The minimum will be halfway between the second two x-intercepts, open parentheses straight pi over 3 comma space 0 close parentheses

and $open parentheses fraction numerator 2 straight pi over denominator 3 end fraction comma space 0 close parentheses comma$ or at x equals straight pi over 2.

This results in a value of y equals short dash 1 half.

The minimum will occur at open parentheses straight pi over 2 comma space short dash 1 half close parentheses.

2c. Vertical Shifts

Recall that the graph of y equals f open parentheses x close parentheses plus d is a vertical shift of d units above the graph of if d greater than 0 and d units below the graph of if d less than 0.

watch

This video will show how the graph of y equals 6 minus 2 cos open parentheses πx close parentheses

relates to the graph of y equals cos x.

big idea

Recall that the graphs of y equals a sin open parentheses b x close parentheses

and

y equals a cos open parentheses b x close parentheses

have midline y equals 0.

When the graph shifts vertically by d units, the midline shifts along with it, and becomes y equals d.

try it

Consider the equation y equals 4 plus 5 sin open parentheses straight pi over 3 x close parentheses.

Identify the period and amplitude.

The period is $fraction numerator 2 straight pi over denominator b end fraction comma$ where b is the coefficient of x. With b equals pi over 3 comma

the period is $fraction numerator 2 straight pi over denominator open parentheses straight pi over 3 close parentheses end fraction equals 6.$

The amplitude is the coefficient of the sine function. Thus, the amplitude is 5.

Identify the points where the graph and the midline intersect and the points where the minimum and maximum values occur on the interval corresponding to the period that starts with x = 0.

Midline Points:

The sine graph intersects its midline at the endpoints of a period as well as the point halfway between the endpoints.
Since the midline of this graph is the line the graph has its y-intercept at With a period of 6, another point on the midline is Then, the point halfway between the endpoints is

Therefore, the graph intersects its midline at open parentheses 0 comma space 4 close parentheses comma

open parentheses 3 comma space 4 close parentheses comma

and

open parentheses 6 comma space 4 close parentheses.

Minimum Point:

The x-coordinate of the minimum point is 4.5, which is halfway between 3 and 6.
The minimum value is which is the position of the midline minus the amplitude.

Therefore, the minimum is located at the point open parentheses 4.5 comma space short dash 1 close parentheses.

Maximum Point:

The x-coordinate of the maximum point is 1.5, which is halfway between 0 and 3.
The maximum value which is the position of the midline added to the amplitude.

Therefore, the maximum is located at the point open parentheses 1.5 comma space 9 close parentheses.

2d. Horizontal (Phase) Shifts

Given the graph of y equals f open parentheses x close parentheses and a positive constant k:

The graph of shifts the graph to the left k units.
The graph of shifts the graph to the right k units.

So far, the y-intercept of a sinusoid corresponds to the minimum value, the midline value, or the maximum value. A horizontal shift could cause the y-intercept to be something other than these three values. When this happens, the horizontal change is called a phase shift.

For example, consider the equations y equals 3 cos open parentheses 2 x close parentheses and y equals 3 cos open square brackets 2 open parentheses x minus straight pi over 6 close parentheses close square brackets.

The graph of y equals 3 cos open square brackets 2 open parentheses x minus straight pi over 6 close parentheses close square brackets is a horizontal translation to the right units from the graph of y equals 3 cos open parentheses 2 x close parentheses.

Note: the function y equals 3 cos open square brackets 2 open parentheses x minus straight pi over 6 close parentheses close square brackets could also be written y equals 3 cos open parentheses 2 x minus straight pi over 3 close parentheses. When there is a coefficient of x in the expression for the angle, it may need to be factored out in order to find the phase shift.

EXAMPLE

Consider the equation y equals 5 minus 4 cos open parentheses 3 x minus straight pi over 4 close parentheses.

Since the coefficient of the sine function is -4, the amplitude is 4 and the graph is reflected over the x-axis. Before the vertical shift, the minimum value is -4 and the maximum value is 4.
Since the constant term is 5, this means the graph is shifted upward 5 units, and its midline is This also means that the minimum and maximum values shift upward 5 units to 1 and 9, respectively.
Since the coefficient of x is 3, the period is $fraction numerator 2 straight pi over denominator 3 end fraction.$
To investigate the horizontal translation, look at the angle,
To determine the phase shift, factor out a 3:
This indicates a phase shift of units to the right.

The graph of y equals 5 minus 4 cos open parentheses 3 x minus straight pi over 4 close parentheses

, along with the midline, y equals 5 comma

are shown below.

Note: another way to find the phase shift is to set the angle equal to 0 and solve for the variable.

In this equation, you have 3 x minus straight pi over 4 equals 0 comma

3 x minus straight pi over 4 equals 0 comma

which gives x equals straight pi over 12.

This result tells you that the point with x-coordinate x equals straight pi over 12

is the new “start” of the curve, which indicates a phase shift of straight pi over 12

units to the right since the original starting place was x equals 0.

In general, given the graph of y equals a sin open parentheses b x minus c close parentheses plus d or y equals a cos open parentheses b x minus c close parentheses plus d comma consider the angle b x minus c. Factoring out b, we obtain b open parentheses x minus c over b close parentheses comma which indicates a phase shift of units.

formula to know

Phase Shift

Given

y equals a sin open parentheses b x minus c close parentheses plus d

y equals a cos open parentheses b x minus c close parentheses plus d comma

the phase shift is c over b

units. A negative value indicates a shift to the left, while a positive value indicates a shift to the right.

try it

Consider the equation y equals 10 minus cos open parentheses 6 πx minus straight pi close parentheses.

Identify the amplitude, period, phase shift, vertical shift, and midline.

The coefficient of the cosine function is -1, which means the amplitude is open vertical bar short dash 1 close vertical bar equals 1.

The coefficient of x is 6 straight pi comma

which means the period is $fraction numerator 2 straight pi over denominator 6 straight pi end fraction equals 1 third.$

To get the phase shift, set the angle 6 straight pi x minus straight pi equals 0.

6 straight pi x minus straight pi equals 0.

Solving for x gives x equals 1 over 6 comma

which means the graph is shifted 1 over 6

units to the right of the basic cosine graph.

Since the constant term is 10, the vertical shift is upward 10 units.

The equation of the midline coincides with the vertical shift; its equation is y equals 10.

Give the values of the minimum and maximum values of the function.

The midline is y equals 10

and the amplitude is 1. This means that the minimum value is 10 minus 1 equals 9

and the maximum value is 10 plus 1 equals 11.

term to know

Phase Shift

The horizontal shift with respect to a sinusoid whose y-intercept is the minimum value, midline value, or maximum value.

3. Writing the Equation of a Sinusoidal Function From Its Graph

Using the aspects of a sinusoidal graph that we discussed earlier, we can write the equation of a sinusoidal function given its graph.

The goal is to write a function in the form y equals a sin open parentheses b x minus c close parentheses plus d or y equals a cos open parentheses b x minus c close parentheses plus d. The goal is to use the most convenient curve, which means avoiding a phase shift, if possible.

If the graph has an extreme point when then use a cosine function.
If the graph intersects its midline when then use a sine function.
If neither of the previous two conditions occur, then there is a phase shift.

EXAMPLE

Write the equation of the function whose graph is shown in the figure below.

Connect information about the function from the properties of the graph:

Property of Graph	Information About Function
The midline of this graph is
The graph passes through the midline at	The model has the form Since the graph passes through its midline at
The function has a maximum value of -1 and a minimum value of -5.	Since the points where the maximum and minimum values occur are two units above and below the midline, the amplitude is 2. Since the graph resembles a non-reflected sine curve,
The period of this function is 4.	$fraction numerator 2 straight pi over denominator b end fraction equals 4 comma$ which means

Substituting the values of a comma

b, c, and d, the equation for the graph is y equals 2 sin open parentheses straight pi over 2 x close parentheses minus 3.

EXAMPLE

Write the equation of the function whose graph is shown in the figure below.

Connect information about the function from the properties of the graph:

Property of Graph	Information About Function
The midline of this graph is
The graph attains its maximum value when	This is a non-reflected cosine graph with no phase shift; therefore, and The model has the form
The function has a maximum value of 2 and a minimum value of -4.	Since these are three units from the midline, the amplitude is 3. Since the graph resembles a non-reflected cosine curve,
The period of this function is	$fraction numerator 2 straight pi over denominator b end fraction equals straight pi comma$ which means

Substituting the values of a comma

b, c, and d, the equation for the graph is y equals 3 cos open parentheses 2 x close parentheses minus 1.

watch

In this video, we’ll use the graph of a sinusoidal function to write its equation.

try it

Consider the graph shown below.

Decide on the most convenient function to use, then write the equation of the function that corresponds to the graph.

The following table is used to link the graph’s properties with the values of a comma

b, c, and d that are used in the equation.

Property of Graph	Information About Function
The midline of this graph is
The graph attains its minimum value when	This is a reflected cosine graph with no phase shift, which means and We seek an equation of the form
The function has a minimum value of 0 and a maximum value of 8.	Each of these values is 4 units from the midline, which means the amplitude is 4. Since we already concluded that is negative,
The period of this function is 6 units.	The period is $fraction numerator 2 straight pi over denominator b end fraction.$ To find b, set $fraction numerator 2 straight pi over denominator b end fraction equals 6.$ Solving, $b equals fraction numerator 2 straight pi over denominator 6 end fraction equals straight pi over 3.$

Substituting a equals short dash 4 comma

and

into the equation y equals a cos open parentheses b x minus c close parentheses plus d comma

the equation of the given graph is y equals 4 minus 4 cos open parentheses straight pi over 3 x close parentheses.

summary

In this lesson, you learned that the graphs of the basic sine and cosine functions are based on the values of the coordinates of the unit circle. Once established, you are able to graph transformations of sine and cosine functions that include reflections, as well as vertical stretches and compressions (amplitude), horizontal stretches and compressions (period), and vertical and horizontal translations. Finally, you learned how to apply this knowledge to write the equation of a sinusoidal function from its graph.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know

Amplitude: The maximum vertical distance that the graph is away from its horizontal midline.
Midline: A horizontal line that represents the value that is the average between the maximum value and the minimum value.
Period of a Function: The smallest value of p for which $f open parentheses x plus p close parentheses equals f open parentheses x close parentheses.$ In other words, if the graph of $f open parentheses x close parentheses$ is shifted horizontally by p units to the left or to the right, the result is the original graph.
Phase Shift: The horizontal shift with respect to a sinusoid whose y-intercept is the minimum value, midline value, or maximum value.
Sinusoidal Graphs (Sinusoids): Any graph that resembles the graph of a sine or cosine function.

Formulas to Know

Period of a Sine or Cosine Function: Assuming $b greater than 0 comma$ the graphs of either $y equals sin open parentheses b x close parentheses$ or $y equals cos open parentheses b x close parentheses$ has period $fraction numerator 2 straight pi over denominator b end fraction.$
Phase Shift: Given $y equals a sin open parentheses b x minus c close parentheses plus d$ or $y equals a cos open parentheses b x minus c close parentheses plus d comma$ the phase shift is $c over b$ units. A negative value indicates a shift to the left, while a positive value indicates a shift to the right.