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8.1 Graphs of the sine and cosine functions  (Page 3/13)

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A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.

Amplitude of sinusoidal functions

If we let   C = 0   and   D = 0   in the general form equations of the sine and cosine functions, we obtain the forms

y = A sin ( B x )  and  y = A cos ( B x )

The amplitude    is   A ,   and the vertical height from the midline    is   | A | .   In addition, notice in the example that

| A |  = amplitude =  1 2 | maximum  −  minimum |

Identifying the amplitude of a sine or cosine function

What is the amplitude of the sinusoidal function   f ( x ) = −4 sin ( x ) ?   Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form   y = A sin ( B x ) .

In the given function,   A = −4 ,   so the amplitude is   | A | = | −4 | = 4.   The function is stretched.

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What is the amplitude of the sinusoidal function   f ( x ) = 1 2 sin ( x ) ?   Is the function stretched or compressed vertically?

1 2   compressed

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Analyzing graphs of variations of y = sin x And y = cos x

Now that we understand how   A   and   B   relate to the general form equation for the sine and cosine functions, we will explore the variables   C   and   D .   Recall the general form:

y = A sin ( B x − C ) + D  and  y = A cos ( B x − C ) + D o r y = A sin ( B ( x − C B ) ) + D  and  y = A cos ( B ( x − C B ) ) + D

The value   C B   for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function    . If   C > 0 ,   the graph shifts to the right. If   C < 0 ,   the graph shifts to the left. The greater the value of   | C | ,   the more the graph is shifted. [link] shows that the graph of   f ( x ) = sin ( x − π )   shifts to the right by   π   units, which is more than we see in the graph of   f ( x ) = sin ( x − π 4 ) ,   which shifts to the right by   π 4   units.

A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi/4), which is the same as sin(x) except shifted to the right by pi/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.

While   C   relates to the horizontal shift,   D   indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function   y = cos ( x ) + D   has its midline at   y = D .

A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.

Any value of   D   other than zero shifts the graph up or down. [link] compares   f ( x ) = sin   x   with   f ( x ) = sin   x + 2 ,   which is shifted 2 units up on a graph.

A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.

Variations of sine and cosine functions

Given an equation in the form   f ( x ) = A sin ( B x − C ) + D   or   f ( x ) = A cos ( B x − C ) + D ,   C B   is the phase shift    and   D   is the vertical shift    .

Identifying the phase shift of a function

Determine the direction and magnitude of the phase shift for   f ( x ) = sin ( x + π 6 ) − 2.

Let’s begin by comparing the equation to the general form   y = A sin ( B x − C ) + D .

In the given equation, notice that   B = 1   and   C = − π 6 .   So the phase shift is

C B = − π 6 1     = − π 6

or   π 6   units to the left.

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Determine the direction and magnitude of the phase shift for   f ( x ) = 3 cos ( x − π 2 ) .

π 2 ;   right

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Identifying the vertical shift of a function

Determine the direction and magnitude of the vertical shift for   f ( x ) = cos ( x ) − 3.

Let’s begin by comparing the equation to the general form   y = A cos ( B x − C ) + D .

In the given equation,   D = −3   so the shift is 3 units downward.

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Determine the direction and magnitude of the vertical shift for   f ( x ) = 3 sin ( x ) + 2.

2 units up

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Given a sinusoidal function in the form   f ( x ) = A sin ( B x − C ) + D ,   identify the midline, amplitude, period, and phase shift.

  1. Determine the amplitude as   | A | .
  2. Determine the period as   P = 2 π | B | .
  3. Determine the phase shift as   C B .
  4. Determine the midline as   y = D .

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
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A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
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8m/s²
Aishat
What is Thermodynamics
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Someone
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Someone
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
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What is specific heat capacity
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Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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