How do I Graph the Sine Function for y=2-sin(2πx/3)?

In summary: I think I understand now.In summary, the standard equation for the given problem is y = A sin(B(x - C)) + D, where the amplitude is A, period is \dfrac{2\pi}{|b|}, phase shift is \dfrac{C}{B}, and vertical shift is D. To generate a table of values, we can use the count formula \dfrac{1}{4}\cdot period to choose our x-values, starting from the phase shift and adding or subtracting \dfrac{3\pi}{4}. The y-values can be found by using the general case formula for sin(x) and substituting in the given values.
  • #1
FritoTaco
132
23

Homework Statement


Problem: [itex]y=2-sin\dfrac{2\pi x}{3}[/itex]

Homework Equations


Standard Equation: [itex] y = A sin(B(x - C)) + D [/itex]
  • A: amplitude is A
  • B: period is [itex]\dfrac{2\pi}{|b|}[/itex]
  • C: phase shift is [itex]\dfrac{C}{B}[/itex]
  • D: vertical shift is D
Count Formula: [itex]\dfrac{1}{4}\cdot period[/itex] (What you use to choose your x-values)

The Attempt at a Solution


First, rearrange the equation: [itex]y=-sin\dfrac{2\pi x}{3}+2[/itex]

Amplitude: [itex]1[/itex] (a-value)
Period: [itex]\dfrac{2\pi}{2\pi/3}=\dfrac{2\pi}{1}\cdot\dfrac{3}{2\pi}=\dfrac{6\pi}{2}=3\pi[/itex]
Phase Shift: [itex]0[/itex] (no c-value)
Vertical Shift: [itex]2[/itex] (d-value)

Table (I don't know how to make):
See my attachment for the table of values.

How did I pick my x-values to calculate my y-values?
I used count formula: [itex]\dfrac{1}{4}\cdot \dfrac{3\pi}{1}=\dfrac{3\pi}{4}[/itex]
So, I start from the phase shift and continue adding [itex]\dfrac{3\pi}{4}[/itex] : [itex]0+\dfrac{3\pi}{4}=\dfrac{3\pi}{4}[/itex]
[itex]\dfrac{3\pi}{4}+\dfrac{3\pi}{4}=\dfrac{6\pi}{4}=\dfrac{3\pi}{2}[/itex]
[itex]\dfrac{3\pi}{2}+\dfrac{3\pi}{4}=\dfrac{6\pi}{4}+\dfrac{3\pi}{4}=\dfrac{9\pi}{4}[/itex]
[itex]\dfrac{9\pi}{4}+\dfrac{3\pi}{4}=\dfrac{12\pi}{4}=3\pi[/itex]

I don't want to graph it yet because my table of values doesn't seem correct, in my book the answer is in my second attachment. They don't even use pi for their x-values :(
 

Attachments

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  • #2
Your period is wrong 3 not 3 pi
All the functions in that family do the same thing
High middle low middle high and so on
We just need one value which state it is then the differences to generate a table
You can add more points too if desired
Vertical change 1 amplitude
Horizontal change .75 period/4
Make a table with horizontal coordinates multiples of 0.75
Multiples of 1.5 have value 2
-3 2
-2.25 1
-1.5 2
-0.75 3
0 2
.75 1
1.5 2
2.25 3
3 2
 
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  • #3
lurflurf said:
Multiples of 1.5 have value 2
-3 2
-2.25 1
-1.5 2
-0.75 3
0 2
.75 1
1.5 2
2.25 3
3 2

Wait, so why are we multiplying are each time by [itex]\dfrac{3}{4}[/itex]? Actually I'm not sure what you are doing. My teacher told me to keep adding it to find new x-values? Where you got 1.5, I got [itex]\dfrac{6}{4}=\dfrac{3}{2}[/itex]
 
Last edited:
  • #4
FritoTaco said:
Where you got 1.5, I got [itex]\dfrac{6}{4}=\dfrac{3}{2}[/itex]
$$\frac 3 2 = 1.5$$
 
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  • #5
But how do the x-values come about? I'm confused.
 
  • #6
start at zero and add or subtract 0.75=3/5
add
0+0.75=0.75
.75+0.75=1.5
1.5+0.75=2.25
2.25+0.75=3
subtract
0-0.75=-0.75
.75-0.75=-1.5
1.5-0.75=-2.25
2.25-0.75=-3

we choose 0.75 because it is p/4=3/4=.075
 
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  • #7
Oh I see, but how did you get the y-values. Like at 0, you have 2 for y value. In my picture I have, I got 1 for my y-value.
 
  • #8
We plot sin(x) for reference using values

$$

\left| \begin{array}{c|c}
x & \sin(x) \\
\hline
-\pi & 0 \\
-\pi/2 & -1\\
0 & 0 \\
\pi/2 & 1\\
\pi & 0\\
\end{array} \right|

$$
We can use more points if we like either to show more periods or more details.

The general case is just shifts and flips and dilation of sin(x)
$$\left| \begin{array}{c|c}
x &A \sin(B(x-C))+D \\
\hline
C-\pi/B & D \\
C-\pi/(2B) & D-A\\
C & D \\
C+\pi/(2B) & D+A\\
C+\pi/B & D\\
\end{array} \right|

$$
including your case
$$\left| \begin{array}{c|c}
x &2- \sin(2\pi x/3) \\
\hline
0-1.5 & 2 \\
0-1.5/2 & 2+1\\
0 & 2 \\
0+1.5/2 & 2-1\\
0+1.5 & 2\\
\end{array} \right|

$$
 
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  • #9
lurflurf said:
including your case
$$\left| \begin{array}{c|c}
x &2- \sin(2\pi x/3) \\
\hline
0-1.5 & 2 \\
0-1.5/2 & 2+1\\
0 & 2 \\
0+1.5/2 & 2-1\\
0+1.5 & 2\\
\end{array} \right|

$$
Oh, I should be including the +2 from my equation. Thanks!
 

What is the sine function?

The sine function is a mathematical function that describes the relationship between the angles of a right triangle and the ratio of the length of its opposite side to its hypotenuse.

What is the equation for the sine function?

The equation for the sine function is y = sin(x), where x is the angle and y is the ratio of the opposite side to the hypotenuse.

How do you graph the sine function?

To graph the sine function, plot points on a coordinate plane by choosing values for x, plugging them into the equation y = sin(x), and then plotting the corresponding y values. Connect the points with a smooth curve to create the graph.

What is the amplitude of the sine function?

The amplitude of the sine function is the distance between the highest point and the lowest point on the graph. It is equal to half the difference between the maximum and minimum values of y in the equation y = sin(x).

What is the period of the sine function?

The period of the sine function is the length of one complete cycle of the graph. It is equal to 2π or 360 degrees, and can be used to determine the frequency of the function.

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