# 6.1 Graphs of the sine and cosine functions  (Page 4/13)

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## Identifying the variations of a sinusoidal function from an equation

Determine the midline, amplitude, period, and phase shift of the function $\text{\hspace{0.17em}}y=3\mathrm{sin}\left(2x\right)+1.$

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

$A=3,\text{\hspace{0.17em}}$ so the amplitude is $\text{\hspace{0.17em}}|A|=3.$

Next, $\text{\hspace{0.17em}}B=2,\text{\hspace{0.17em}}$ so the period is $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}=\frac{2\pi }{2}=\pi .$

There is no added constant inside the parentheses, so $\text{\hspace{0.17em}}C=0\text{\hspace{0.17em}}$ and the phase shift is $\text{\hspace{0.17em}}\frac{C}{B}=\frac{0}{2}=0.$

Finally, $\text{\hspace{0.17em}}D=1,\text{\hspace{0.17em}}$ so the midline is $\text{\hspace{0.17em}}y=1.$

Determine the midline, amplitude, period, and phase shift of the function $\text{\hspace{0.17em}}y=\frac{1}{2}\mathrm{cos}\left(\frac{x}{3}-\frac{\pi }{3}\right).$

midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ amplitude: $\text{\hspace{0.17em}}|A|=\frac{1}{2};\text{\hspace{0.17em}}$ period: $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}=6\pi ;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}\frac{C}{B}=\pi$

## Identifying the equation for a sinusoidal function from a graph

Determine the formula for the cosine function in [link] .

To determine the equation, we need to identify each value in the general form of a sinusoidal function.

$\begin{array}{l}y=A\mathrm{sin}\left(Bx-C\right)+D\hfill \\ y=A\mathrm{cos}\left(Bx-C\right)+D\hfill \end{array}$

The graph could represent either a sine or a cosine function    that is shifted and/or reflected. When $\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ the graph has an extreme point, $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ Since the cosine function has an extreme point for $\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ let us write our equation in terms of a cosine function.

Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below $\text{\hspace{0.17em}}y=0.5.\text{\hspace{0.17em}}$ This value, which is the midline, is $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ in the equation, so $\text{\hspace{0.17em}}D=0.5.$

The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So $\text{\hspace{0.17em}}|A|=0.5.\text{\hspace{0.17em}}$ Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so $\text{\hspace{0.17em}}|A|=\frac{1}{2}=0.5.\text{\hspace{0.17em}}$ Also, the graph is reflected about the x -axis so that $\text{\hspace{0.17em}}A=-0.5.$

The graph is not horizontally stretched or compressed, so $\text{\hspace{0.17em}}B=1;\text{\hspace{0.17em}}$ and the graph is not shifted horizontally, so $\text{\hspace{0.17em}}C=0.$

Putting this all together,

$g\left(x\right)=-0.5\mathrm{cos}\left(x\right)+0.5$

Determine the formula for the sine function in [link] .

$f\left(x\right)=\mathrm{sin}\left(x\right)+2$

## Identifying the equation for a sinusoidal function from a graph

Determine the equation for the sinusoidal function in [link] .

With the highest value at 1 and the lowest value at $\text{\hspace{0.17em}}-5,\text{\hspace{0.17em}}$ the midline will be halfway between at $\text{\hspace{0.17em}}-2.\text{\hspace{0.17em}}$ So $\text{\hspace{0.17em}}D=-2.\text{\hspace{0.17em}}$

The distance from the midline to the highest or lowest value gives an amplitude of $\text{\hspace{0.17em}}|A|=3.$

The period of the graph is 6, which can be measured from the peak at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ to the next peak at $\text{\hspace{0.17em}}x=7,$ or from the distance between the lowest points. Therefore, $P=\frac{2\pi }{|B|}=6.\text{\hspace{0.17em}}$ Using the positive value for $\text{\hspace{0.17em}}B,$ we find that

$B=\frac{2\pi }{P}=\frac{2\pi }{6}=\frac{\pi }{3}$

So far, our equation is either $\text{\hspace{0.17em}}y=3\mathrm{sin}\left(\frac{\pi }{3}x-C\right)-2\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}y=3\mathrm{cos}\left(\frac{\pi }{3}x-C\right)-2.\text{\hspace{0.17em}}$ For the shape and shift, we have more than one option. We could write this as any one of the following:

• a cosine shifted to the right
• a negative cosine shifted to the left
• a sine shifted to the left
• a negative sine shifted to the right

While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes

Again, these functions are equivalent, so both yield the same graph.

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
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
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
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?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
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.
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?
what is foci?
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