# 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

give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena