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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 golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
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what is viscosity?
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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