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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

foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
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?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
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
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
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
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
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
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
Practice Key Terms 5

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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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