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

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## Amplitude of sinusoidal functions

If we let $\text{\hspace{0.17em}}C=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D=0\text{\hspace{0.17em}}$ in the general form equations of the sine and cosine functions, we obtain the forms

The amplitude    is $\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ and the vertical height from the midline    is $\text{\hspace{0.17em}}|A|.\text{\hspace{0.17em}}$ In addition, notice in the example that

## Identifying the amplitude of a sine or cosine function

What is the amplitude of the sinusoidal function $\text{\hspace{0.17em}}f\left(x\right)=-4\mathrm{sin}\left(x\right)?\text{\hspace{0.17em}}$ Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form $\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx\right).$

In the given function, $\text{\hspace{0.17em}}A=-4,\text{\hspace{0.17em}}$ so the amplitude is $\text{\hspace{0.17em}}|A|=|-4|=4.\text{\hspace{0.17em}}$ The function is stretched.

What is the amplitude of the sinusoidal function $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}\mathrm{sin}\left(x\right)?\text{\hspace{0.17em}}$ Is the function stretched or compressed vertically?

$\frac{1}{2}\text{\hspace{0.17em}}$ compressed

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

Now that we understand how $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ relate to the general form equation for the sine and cosine functions, we will explore the variables $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D.\text{\hspace{0.17em}}$ Recall the general form:

The value $\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function    . If $\text{\hspace{0.17em}}C>0,\text{\hspace{0.17em}}$ the graph shifts to the right. If $\text{\hspace{0.17em}}C<0,\text{\hspace{0.17em}}$ the graph shifts to the left. The greater the value of $\text{\hspace{0.17em}}|C|,\text{\hspace{0.17em}}$ the more the graph is shifted. [link] shows that the graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x-\pi \right)\text{\hspace{0.17em}}$ shifts to the right by $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ units, which is more than we see in the graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x-\frac{\pi }{4}\right),\text{\hspace{0.17em}}$ which shifts to the right by $\text{\hspace{0.17em}}\frac{\pi }{4}\text{\hspace{0.17em}}$ units.

While $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ relates to the horizontal shift, $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function $\text{\hspace{0.17em}}y=\mathrm{cos}\left(x\right)+D\text{\hspace{0.17em}}$ has its midline at $\text{\hspace{0.17em}}y=D.$

Any value of $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ other than zero shifts the graph up or down. [link] compares $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x+2,\text{\hspace{0.17em}}$ which is shifted 2 units up on a graph.

## Variations of sine and cosine functions

Given an equation in the form $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sin}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cos}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ $\frac{C}{B}\text{\hspace{0.17em}}$ is the phase shift    and $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ is the vertical shift    .

## Identifying the phase shift of a function

Determine the direction and magnitude of the phase shift for $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x+\frac{\pi }{6}\right)-2.$

Let’s begin by comparing the equation to the general form $\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx-C\right)+D.$

In the given equation, notice that $\text{\hspace{0.17em}}B=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=-\frac{\pi }{6}.\text{\hspace{0.17em}}$ So the phase shift is

or $\text{\hspace{0.17em}}\frac{\pi }{6}\text{\hspace{0.17em}}$ units to the left.

Determine the direction and magnitude of the phase shift for $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{cos}\left(x-\frac{\pi }{2}\right).$

$\frac{\pi }{2};\text{\hspace{0.17em}}$ right

## Identifying the vertical shift of a function

Determine the direction and magnitude of the vertical shift for $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\left(x\right)-3.$

Let’s begin by comparing the equation to the general form $\text{\hspace{0.17em}}y=A\mathrm{cos}\left(Bx-C\right)+D.$

In the given equation, $\text{\hspace{0.17em}}D=-3\text{\hspace{0.17em}}$ so the shift is 3 units downward.

Determine the direction and magnitude of the vertical shift for $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{sin}\left(x\right)+2.$

2 units up

Given a sinusoidal function in the form $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sin}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ identify the midline, amplitude, period, and phase shift.

1. Determine the amplitude as $\text{\hspace{0.17em}}|A|.$
2. Determine the period as $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.$
3. Determine the phase shift as $\text{\hspace{0.17em}}\frac{C}{B}.$
4. Determine the midline as $\text{\hspace{0.17em}}y=D.$

#### Questions & Answers

A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply

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