# 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

"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo