Graph variations of y=sin( x ) and also y=cos( x )

Recall that the sine and also cosine functions relate actual number worths to the x– and also y-coordinates the a point on the unit circle. So what perform they look favor on a graph on a name: coordinates plane? Let’s start with the sine function.

You are watching: Using complete sentences, explain the key features of the graph of the cosine function.

we can develop a table the values and use castle to map out a graph. The table below lists some of the worths for the sine duty on a unit circle.

x0\fracπ6\\\fracπ6\\\fracπ3\\\fracπ2\\\frac2π3\\\frac3π4\\\frac5π6\\π
sin(x)0

\frac12\\

\frac\sqrt22\\

\frac\sqrt32\\

1\frac\sqrt32\\\frac\sqrt22\\\frac12\\0

Plotting the points indigenous the table and also continuing along the x-axis provides the form of the sine function. See figure 2.


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Figure 3. Plotting worths of the sine function


Now let’s take a similar look in ~ the cosine function. Again, we can produce a table that values and also use castle to lay out a graph. The table below lists some of the worths for the cosine function on a unit circle.

x0\fracπ6\\\fracπ4\\\fracπ3\\\fracπ2\\\frac2π3\\\frac3π4\\\frac5π6\\π
cos(x)1\frac\sqrt32\\\frac\sqrt22\\\frac12\\
0
−\frac12\\
−\frac\sqrt22\\−\frac\sqrt32\\−1

As with the sine function, we deserve to plots point out to develop a graph the the cosine duty as in Figure 4.


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Figure 5


Looking again at the sine and cosine features on a domain focused at the y-axis helps disclose symmetries. Together we can see in number 6, the sine function is symmetric around the origin. Recall indigenous The various other Trigonometric attributes that we established from the unit circle the the sine duty is one odd duty because sin(−x)=−sinx. Currently we can clearly see this residential or commercial property from the graph.


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Figure 7. also symmetry that the cosine function


A general Note: characteristics of Sine and also Cosine Functions

The sine and also cosine attributes have several unique characteristics:

They room periodic attributes with a period of 2π.The domain the each role is (−∞,∞) and the selection is <−1,1>.The graph that y = sin x is symmetric about the origin, due to the fact that it is one odd function.The graph of y = cos x is symmetric about the y-axis, because it is an even function.

Investigating Sinusoidal Functions

As we deserve to see, sine and also cosine features have a regular period and range. If we watch ocean waves or ripples top top a pond, us will view that they resemble the sine or cosine functions. However, they space not necessarily identical. Some are taller or longer than others. A role that has actually the same basic shape together a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal features are


y = A sin (BxC) + D

and


y = A cos (Bx − C) + D

Determining the period of Sinusoidal Functions

Looking in ~ the creates of sinusoidal functions, we have the right to see that they are changes of the sine and also cosine functions. We have the right to use what we know around transformations to identify the period.

In the basic formula, B is regarded the duration by \textP = \frac2πB. If |B| > 1, then the period is much less than 2π and also the role undergoes a horizontal compression, vice versa, if |B|

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Figure 9


A general Note: Amplitude of Sinusoidal Functions

If us let = 0 and also D = 0 in the general kind equations the the sine and cosine functions, we attain the forms


y=A\sin(Bx)\\ and y=A\cos(Bx)\\

The amplitude is A, and the vertical elevation from the midline is|A|. In addition, an alert in the example that


|A|=\textamplitude=\frac12|\textmaximum−\textminimum|\\

Example 2: identify the Amplitude of a Sine or Cosine Function

What is the amplitude of the sinusoidal function f(x)=−4\sin(x)\\? Is the role stretched or compressed vertically?


Solution

Let’s begin by to compare the function to the simplified type y = A sin(Bx).

In the offered function, = −4, so the amplitude is |A|=|−4| = 4. The function is stretched.


Analysis the the Solution

The negative value of A results in a reflection across the x-axis of the sine function, as presented in figure 10.


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Figure 11


While C relates come the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. The function y=\cos(x)+D\\ has actually its midline at y=D.


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Figure 13


A general Note: sports of Sine and Cosine Functions

Given one equation in the form f(x)=A\sin(Bx−C)+D\\ or f(x)=A\cos(Bx−C)+D\\, \fracCB\\is the phase shift and D is the vertical shift.


Example 3: identifying the Phase change of a Function

Determine the direction and also magnitude that the phase shift for f(x)=\sin(x+\fracπ6)−2\\.


Solution

Let’s start by compare the equation to the general form y=A\sin(Bx−C)+D\\.

In the provided equation, notification that B = 1 and C=−\fracπ6\\. So the phase shift is

\beginarray\fracCB=−\frac\fracx61\hfill \\ =−\frac\pi6\hfill \endarray\\

or \frac\pi6\\ units to the left.


Analysis of the Solution

We need to pay attention to the authorize in the equation because that the general type of a sinusoidal function. The equation mirrors a minus sign prior to C. As such f(x)=\sin(x+\fracπ6)−2\\ can be rewritten together f(x)=\sin(x−(−\fracπ6))−2\\. If the value of C is negative, the shift is come the left.


Try the 3

Determine the direction and magnitude of the phase transition for f(x)=3\cos(x−\frac\pi2)\\.

Solution


Example 4: identify the Vertical change of a Function

Determine the direction and also magnitude that the vertical shift for f(x)=\cos(x)−3\\.


Solution

Let’s start by comparing the equation come the general kind y=A\cos(Bx−C)+D\\


Try it 4

Determine the direction and magnitude of the vertical change for f(x)=3\sin(x)+2\\.

Solution


How To: offered a sinusoidal duty in the kind f(x)=A\sin(Bx−C)+D, identify the midline, amplitude, period, and also phase shift.

Determine the amplitude as|A|.Determine the period as P=\frac2π\\.Determine the phase shift as \fracCB\\.Determine the midline as = D.

Example 5: identifying the sports of a Sinusoidal function from an Equation

Determine the midline, amplitude, period, and also phase shift of the role y=3\sin(2x)+1\\.


Solution

Let’s start by compare the equation to the general kind y=A\sin(Bx−C)+D\\A = 3, so the amplitude is |A| = 3.

Next, B = 2, therefore the period is \textP=\frac2π=\frac2π2=π\\.

There is no added continuous inside the parentheses, so C = 0 and the phase transition is \fracCB=\frac02=0\\.

Finally, D = 1, therefore the midline is y = 1.


Analysis that the Solution

Inspecting the graph, we have the right to determine the the period is π, the midline is y = 1,and the amplitude is 3. See number 14.


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Figure 15


Solution


f(x)=\sin(x)+2\\

Try the 6

Determine the formula because that the sine duty in figure 16.


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Figure 17


Solution

With the highest possible value in ~ 1 and the lowest value at−5, the midline will be halfway in between at −2. Therefore D = −2.

The street from the midline come the greatest or shortest value provides an amplitude that |A|=3.

The period of the graph is 6, which have the right to be measured native the peak at = 1 come the following peak in ~ x = 7, or from the distance in between the lowest points. Therefore, \textP=\frac2\pi=6. Using the confident value for B, we uncover that

B=\frac2πP=\frac2π6=\fracπ3\\

So far, ours equation is either y=3\sin(\frac\pi3x−C)−2\\ or y=3\cos(\frac\pi3x−C)−2\\. For the shape and shift, we have more than one option. We can write this as any one the the following:

a cosine change to the righta an adverse cosine shifted to the lefta sine shifted to the lefta negative sine shifted to the right

While any kind of of these would be correct, the cosine shifts are much easier to occupational with 보다 the sine move in this case due to the fact that they involve creature values. Therefore our function becomes

y=3\cos(\fracπ3x−\fracπ3)−2\\ or y=−3\cos(\fracπ3x+\frac2π3)−2\\

Again, these attributes are equivalent, for this reason both productivity the very same graph.


Try the 7

Write a formula for the function graphed in figure 18.


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Figure 19


Try that 8

Sketch a graph of g(x)=−0.8\cos(2x)\\. Recognize the midline, amplitude, period, and phase shift.

Solution


How To: given a sinusoidal role with a phase shift and a upright shift, sketch its graph.

Express the function in the general kind y=A\sin(Bx−C)+D or y=A\cos(Bx−C)+D\\.Identify the amplitude, |A|.Identify the period, P=2π|B|.Identify the phase shift, \fracCB\\.Draw the graph the f(x)=A\sin(Bx)\\ shifted to the appropriate or left by \fracCB\\ and also up or under by D.

Example 9: Graphing a changed Sinusoid

Sketch a graph of f(x)=3\sin(\fracπ4x−\fracπ4)\\.


Solution

Step 1. The function is currently written in basic form: f(x)=3\sin(\fracπ4x−\fracπ4)\\. This graph will have actually the shape of a sine function, beginning at the midline and also increasing come the right.

Step 2. |A|=|3|=3. The amplitude is 3.

Step 3. since |B|=|\fracπ4|=\fracπ4\\, we recognize the duration as follows.

\textP=\frac2πB=\frac2π\fracπ4=2π\times\frac4π=8\\

The duration is 8.

Step 4. because \textC=\fracπ4\\, the phase shift is

\fracCB=\frac\frac\pi4\frac\pi4=1\\.

The phase change is 1 unit.

Step 5. number 20 reflects the graph the the function.


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Figure 21


Using revolutions of Sine and Cosine Functions

We deserve to use the changes of sine and cosine features in countless applications. As discussed at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.


Example 11: detect the vertical Component of one Motion

A suggest rotates approximately a one of radius 3 focused at the origin. Map out a graph the the y-coordinate the the point as a function of the edge of rotation.


Solution

Recall that, for a suggest on a circle of radius r, the y-coordinate of the allude is y=r\sin(x), therefore in this case, we get the equation y(x)=3\sin(x). The consistent 3 causes a vertical stretch of the y-values the the role by a variable of 3, i m sorry we have the right to see in the graph in number 22.


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Figure 23


Solution

Sketching the height, we keep in mind that that will begin 1 ft above the ground, then boost up to 7 ft above the ground, and continue come oscillate 3 ft above and listed below the center value of 4 ft, as presented in number 24.


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Figure 25


Solution


Example 13: determining a Rider’s height on a Ferris Wheel

The London Eye is a substantial Ferris wheel v a diameter the 135 meters (443 feet). That completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. To express a rider’s height over ground as a role of time in minutes.


Solution

With a diameter the 135 m, the wheel has a radius of 67.5 m. The elevation will oscillate through amplitude 67.5 m over and listed below the center.

Passengers plank 2 m above ground level, so the center of the wheel have to be situated 67.5 + 2 = 69.5 m over ground level. The midline the the oscillation will certainly be in ~ 69.5 m.

The wheel take away 30 minutes to complete 1 revolution, therefore the elevation will oscillate v a period of 30 minutes.

Lastly, due to the fact that the rider boards in ~ the shortest point, the elevation will start at the the smallest value and also increase, following the shape of a vertically reflect cosine curve.

Amplitude: 67.5, so = 67.5Midline: 69.5, for this reason D = 69.5Period: 30, therefore B=\frac2\pi30=\frac\pi15Shape: −cos(t)

An equation for the rider’s height would be

y=−67.5\cos(\frac\pi15t)+69.5

where t is in minutes and y is measure up in meters.


Key Equations

Sinusoidal functionsf(x)=A\sin(Bx−C)+D
f(x)=A\cos(Bx−C)+D
Key ConceptsPeriodic functions repeat ~ a given value. The smallest such worth is the period. The simple sine and cosine functions have a duration of 2π.The function sin x is odd, for this reason its graph is symmetric about the origin. The function cos x is even, therefore its graph is symmetric around the y-axis.The graph of a sinusoidal function has the same basic shape together a sine or cosine function.In the general formula because that a sinusoidal function, the period is \textP=\frac2\piB.In the basic formula because that a sinusoidal function, |A|represents amplitude. If |A| > 1, the function is stretched, whereas if|A| The worth \fracCB in the basic formula because that a sinusoidal role indicates the step shift.The value D in the general formula for a sinusoidal duty indicates the vertical transition from the midline.Combinations of sport of sinusoidal functions can be detected indigenous an equation.The equation because that a sinusoidal role can be established from a graph.A duty can be graphed by identifying its amplitude and also period.A function can additionally be graphed by identifying its amplitude, period, phase shift, and horizontal shift.Sinusoidal features can be supplied to resolve real-world problems.

Glossary

amplitudethe vertical height of a function; the consistent A showing up in the definition of a sinusoidal functionmidlinethe horizontal line = D, whereby D appears in the general form of a sinusoidal functionperiodic functiona role f(x) the satisfies f(x+P)=f(x) because that a specific constant and any type of value of xphase shiftthe horizontal displacement the the simple sine or cosine function; the continuous \fracCBsinusoidal functionany duty that deserve to be expressed in the kind f(x)=A\sin(Bx−C)+D or f(x)=A\cos(Bx−C)+D

Section Exercises

1. Why room the sine and also cosine functions called periodic functions?

2. How does the graph the y=\sin x compare through the graph the y=\cos x? describe how you could horizontally analyze the graph that y=\sin x to obtain y=\cos x.

3. For the equation A\cos(Bx+C)+D, what constants affect the range of the function and how do they affect the range?

4. Exactly how does the range of a translated sine duty relate come the equation y=A\sin(Bx+C)+D?

5. How deserve to the unit circle be supplied to construct the graph the f(t)=\sin t?

6. f(x)=2\sin x

7. f(x)=\frac23\cos x

8. f(x)=−3\sin x

9. f(x)=4\sin x

10. f(x)=2\cos x

11. f(x)=\cos(2x)

12. f(x)=2\sin(\frac12x)

13. f(x)=4\cos(\pi x)

14. f(x)=3\cos(\frac65x)

15. y=3\sin(8(x+4))+5

16. y=2\sin(3x−21)+4

17. y=5\sin(5x+20)−2

For the adhering to exercises, graph one full period of each function, beginning at x=0. For each function, state the amplitude, period, and midline. State the maximum and also minimum y-values and their equivalent x-values top top one period for x>0. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

18. f(t)=2\sin(t−\frac5\pi6)

19. f(t)=−\cos(t+\frac\pi3)+1

20. f(t)=4\cos(2(t+\frac\pi4))−3

21. f(t)=−\sin(12t+\frac5\pi3)

22. f(x)=4\sin(\frac\pi2(x−3))+7

23. Determine the amplitude, midline, period, and also an equation including the sine duty for the graph displayed in figure 26.


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Figure 27


25. Identify the amplitude, period, midline, and an equation entailing cosine for the graph shown in figure 28.


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Figure 29


27. Identify the amplitude, period, midline, and an equation involving cosine for the graph shown in figure 30.


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Figure 30


28. Identify the amplitude, period, midline, and also an equation including sine for the graph displayed in number 31.


29. Determine the amplitude, period, midline, and also an equation including cosine for the graph displayed in figure 32.


30. Determine the amplitude, period, midline, and an equation involving sine because that the graph presented in figure 33.


For the complying with exercises, let f(x)=\sin x.

31. On <0,2π), fix f(x)=\frac12.

32. Evaluate f(\frac\pi2).

33. ~ above <0,2π), f(x)=\frac\sqrt22. Uncover all worths of x.

34. On <0,2π), the best value(s) that the role occur(s) in ~ what x-value(s)?

35. ~ above <0,2π), the minimum value(s) the the role occur(s) at what x-value(s)?

36. Present that f(−x)=−f(x).This means that f(x)=\sin x is one odd duty and own symmetry with respect come ________________.

For the following exercises, permit f(x)=\cos x.

37. On <0,2π), resolve the equation f(x)=\cos x=0.

38. On<0,2π), solve f(x)=\frac12.

39. ~ above <0,2π), discover the x-intercepts the f(x)=\cos x.

40. ~ above <0,2π), discover the x-values in ~ which the role has a maximum or minimum value.

41. Top top <0,2π), resolve the equation f(x)=\frac\sqrt32.

42. Graph h(x)=x+\sin x \text on<0,2\pi>. Define why the graph shows up as that does.

43. Graph h(x)=x+\sin x on<−100,100>. Did the graph appear as predicted in the vault exercise?

44. Graph f(x)=x\sin x top top <0,2π> and also verbalize how the graph varies from the graph the f(x)=\sin x.

45. Graph f(x)=x\sin x on the window <−10,10> and also explain what the graph shows.

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46. Graph f(x)=\frac\sin xx top top the window <−5π,5π> and explain what the graph shows.

47. A Ferris wheel is 25 meter in diameter and boarded indigenous a platform that is 1 meter over the ground. The six o’clock position on the Ferris wheel is level through the loading platform. The wheel completes 1 full transformation in 10 minutes. The duty h(t) offers a person’s elevation in meters above the ground t minutes after the wheel starts to turn.a. Uncover the amplitude, midline, and duration of h(t).b. Uncover a formula because that the height function h(t).c. Just how high turn off the soil is a human being after 5 minutes?