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.

Questions & Answers

x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
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
Carlos Reply
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
Brad
How can you tell what type of parent function a graph is ?
Mary Reply
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)=
Karim Reply
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
GREAT ANSWER THOUGH!!!
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?
unknown Reply
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
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
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.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
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
Jhon Reply
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
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake

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