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For the equation A cos ( B x + C ) + D , what constants affect the range of the function and how do they affect the range?

The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically.

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How does the range of a translated sine function relate to the equation y = A sin ( B x + C ) + D ?

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How can the unit circle be used to construct the graph of f ( t ) = sin t ?

At the point where the terminal side of t intersects the unit circle, you can determine that the sin t equals the y -coordinate of the point.

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Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for x > 0. Round answers to two decimal places if necessary.

f ( x ) = 2 3 cos x

A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].

amplitude: 2 3 ; period: 2 π ; midline: y = 0 ; maximum: y = 2 3 occurs at x = 0 ; minimum: y = 2 3 occurs at x = π ; for one period, the graph starts at 0 and ends at 2 π

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f ( x ) = 3 sin x

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f ( x ) = 4 sin x

A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].

amplitude: 4; period: 2 π ; midline: y = 0 ; maximum y = 4 occurs at x = π 2 ; minimum: y = 4 occurs at x = 3 π 2 ; one full period occurs from x = 0 to x = 2 π

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f ( x ) = cos ( 2 x )

A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].

amplitude: 1; period: π ; midline: y = 0 ; maximum: y = 1 occurs at x = π ; minimum: y = 1 occurs at x = π 2 ; one full period is graphed from x = 0 to x = π

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f ( x ) = 2 sin ( 1 2 x )

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f ( x ) = 4 cos ( π x )

A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].

amplitude: 4; period: 2; midline: y = 0 ; maximum: y = 4 occurs at x = 0 ; minimum: y = 4 occurs at x = 1

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f ( x ) = 3 cos ( 6 5 x )

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y = 3 sin ( 8 ( x + 4 ) ) + 5

A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.

amplitude: 3; period: π 4 ; midline: y = 5 ; maximum: y = 8 occurs at x = 0.12 ; minimum: y = 2 occurs at x = 0.516 ; horizontal shift: 4 ; vertical translation 5; one period occurs from x = 0 to x = π 4

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y = 2 sin ( 3 x 21 ) + 4

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y = 5 sin ( 5 x + 20 ) 2

A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].

amplitude: 5; period: 2 π 5 ; midline: y = −2 ; maximum: y = 3 occurs at x = 0.08 ; minimum: y = −7 occurs at x = 0.71; phase shift: −4 ; vertical translation: −2; one full period can be graphed on x = 0 to x = 2 π 5

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

f ( t ) = 2 sin ( t 5 π 6 )

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f ( t ) = cos ( t + π 3 ) + 1

A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.

amplitude: 1 ; period: 2 π ; midline: y = 1 ; maximum: y = 2 occurs at x = 2.09 ; maximum: y = 2 occurs at t = 2.09 ; minimum: y = 0 occurs at t = 5.24 ; phase shift: π 3 ; vertical translation: 1; one full period is from t = 0 to t = 2 π

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f ( t ) = 4 cos ( 2 ( t + π 4 ) ) 3

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f ( t ) = sin ( 1 2 t + 5 π 3 )

A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3.

amplitude: 1; period: 4 π ; midline: y = 0 ; maximum: y = 1 occurs at t = 11.52 ; minimum: y = 1 occurs at t = 5.24 ; phase shift: 10 π 3 ; vertical shift: 0

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f ( x ) = 4 sin ( π 2 ( x 3 ) ) + 7

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Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in [link] .

A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.

amplitude: 2; midline: y = 3 ; period: 4; equation: f ( x ) = 2 sin ( π 2 x ) 3

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Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

A graph with a cosine parent function, with amplitude of 3, period of pi, midline at y=-1, and range of [-4,2]
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Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

A graph with a cosine parent function with an amplitude of 2, period of 5, midline at y=3, and a range of [1,5].

amplitude: 2; period: 5; midline: y = 3 ; equation: f ( x ) = 2 cos ( 2 π 5 x ) + 3

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Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .

A sinusoidal graph with amplitude of 4, period of 10, midline at y=0, and range [-4,4].
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Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.

amplitude: 4; period: 2; midline: y = 0 ; equation: f ( x ) = 4 cos ( π ( x π 2 ) )

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Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .

A graph with sine parent function. Amplitude 2, period 2, midline y=0
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Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

A graph with cosine parent function. Amplitude 2, period 2, midline y=1

amplitude: 2; period: 2; midline y = 1 ; equation: f ( x ) = 2 cos ( π x ) + 1

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Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .

A graph with a sine parent function. Amplitude 1, period 4 and midline y=0.
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Algebraic

For the following exercises, let f ( x ) = sin x .

On [ 0 , 2 π ), solve f ( x ) = 0.

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On [ 0 , 2 π ), solve f ( x ) = 1 2 .

π 6 , 5 π 6

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Evaluate f ( π 2 ) .

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On [ 0 , 2 π ) , f ( x ) = 2 2 . Find all values of x .

π 4 , 3 π 4

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On [ 0 , 2 π ), the maximum value(s) of the function occur(s) at what x -value(s)?

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On [ 0 , 2 π ), the minimum value(s) of the function occur(s) at what x -value(s)?

3 π 2

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Show that f ( x ) = f ( x ) . This means that f ( x ) = sin x is an odd function and possesses symmetry with respect to ________________.

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

On [ 0 , 2 π ), solve the equation f ( x ) = cos x = 0.

π 2 , 3 π 2

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On [ 0 , 2 π ), solve f ( x ) = 1 2 .

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On [ 0 , 2 π ), find the x -intercepts of f ( x ) = cos x .

π 2 , 3 π 2

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On [ 0 , 2 π ), find the x -values at which the function has a maximum or minimum value.

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On [ 0 , 2 π ), solve the equation f ( x ) = 3 2 .

π 6 , 11 π 6

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Technology

Graph h ( x ) = x + sin x on [ 0 , 2 π ] . Explain why the graph appears as it does.

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Graph h ( x ) = x + sin x on [ 100 , 100 ] . Did the graph appear as predicted in the previous exercise?

The graph appears linear. The linear functions dominate the shape of the graph for large values of x .

A sinusoidal graph that increases like the function y=x, shown from 0 to 100.
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Graph f ( x ) = x sin x on [ 0 , 2 π ] and verbalize how the graph varies from the graph of f ( x ) = sin x .

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Graph f ( x ) = x sin x on the window [ −10 , 10 ] and explain what the graph shows.

The graph is symmetric with respect to the y -axis and there is no amplitude because the function is not periodic.

A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.
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Graph f ( x ) = sin x x on the window [ −5 π , 5 π ] and explain what the graph shows.

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Real-world applications

A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h ( t ) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.

  1. Find the amplitude, midline, and period of h ( t ) .
  2. Find a formula for the height function h ( t ) .
  3. How high off the ground is a person after 5 minutes?
  1. Amplitude: 12.5; period: 10; midline: y = 13.5 ;
  2. h ( t ) = 12.5 sin ( π 5 ( t 2.5 ) ) + 13.5 ;
  3. 26 ft
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Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
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Sin(A+B) = sinBcosA+cosBsinA
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Prove it
Eseka
Please prove it
Eseka
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Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
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I think you should say "28 terms" instead of "28th term"
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the 28th term is 175
Nando
192
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if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
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Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
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Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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