# 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.

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
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil