Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range

The Organic Chemistry Tutor
22 Feb 201618:34
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial provides a comprehensive guide on graphing sine and cosine functions, focusing on the impact of horizontal phase shifts. It begins with basic equations like y=sin(x) and y=cos(x), explaining amplitude, period, and how to plot them. The video then progresses to more complex functions, detailing how to handle amplitude changes, vertical and horizontal shifts, and period adjustments. Examples are provided to illustrate the process of graphing functions with different parameters, such as y=2sin(x), y=3cos(1/3x), and y=-2sin(1/3x)+3, among others. The tutorial concludes by emphasizing the domain and range considerations for these trigonometric functions.

Takeaways
  • πŸ“ˆ The basic equation for sine and cosine functions is y = a * sin(Bx + C) + D, where 'a' is the amplitude, 'B' affects the period, 'C' indicates horizontal shifts, and 'D' is the vertical shift.
  • 🌟 The amplitude ('a') determines the range of the function between its maximum and minimum values, typically between 1 and -1 for the basic sine and cosine functions.
  • πŸ”„ The period of a sine or cosine function is given by 2Ο€/B, where 'B' is the coefficient of 'x'. The period represents the length of one complete cycle of the function.
  • πŸ“‰ For the basic sine function (y = sin(x)), the period is 2Ο€, and it starts at the center, goes up, down, and returns to the center, repeating this pattern.
  • πŸ“Š Negative sine (y = -sin(x)) inverts the function, so it goes down first instead of up, and the graph is a reflection across the x-axis.
  • πŸ“ˆ The cosine function (y = cos(x)) starts at the top, goes down, up, and returns to the center, with a period of 2Ο€ like the sine function.
  • πŸ”„ When graphing y = 2sin(x), the vertical stretch is applied, doubling the amplitude to 2 and halving the period to Ο€.
  • πŸ”„ For y = 3cos(1/3x), the amplitude is 3, and there is a horizontal stretch by a factor of 3, with a period of 9Ο€.
  • πŸ”„ The vertical shift ('D') is added to or subtracted from the function's value to adjust the graph up or down.
  • πŸ”„ The phase shift ('C') is calculated by setting the function equal to zero and solving for 'x'; this determines where the graph starts.
  • πŸ” The domain for sine and cosine graphs is typically from negative infinity to infinity unless restricted, and the range varies depending on the amplitude and vertical shift.
Q & A

  • What is the basic equation for a sine function with amplitude and period?

    -The basic equation for a sine function is y = sin(x), where the amplitude is 1 and the period is 2Ο€.

  • How does the graph of y = -sin(x) differ from y = sin(x)?

    -The graph of y = -sin(x) is a reflection of y = sin(x) across the x-axis, meaning it starts by going down instead of up.

  • What is the starting point of the cosine function graph?

    -The starting point of the cosine function graph is at the top, unlike the sine function which starts at the center.

  • How does the equation y = 2sin(x) differ from y = sin(x) in terms of amplitude?

    -The equation y = 2sin(x) has an amplitude of 2, meaning it varies between 2 and -2 instead of 1 and -1 as in y = sin(x).

  • What is the period of the function y = sin(2x)?

    -The period of the function y = sin(2x) is Ο€, which is half the period of y = sin(x).

  • What does the coefficient 'a' represent in the general equation for sine or cosine waves, y = a * sin(Bx + C) + D?

    -In the general equation y = a * sin(Bx + C) + D, 'a' represents the amplitude of the function.

  • What does the coefficient 'B' in the general equation determine?

    -The coefficient 'B' in the general equation determines the period of the function. The period is calculated as 2Ο€/B.

  • What does the coefficient 'C' in the general equation indicate?

    -The coefficient 'C' in the general equation indicates the presence of any horizontal phase shift.

  • What is the domain of sine and cosine functions when no restriction is applied?

    -The domain of sine and cosine functions when no restriction is applied is from negative infinity to infinity.

  • How do you calculate the period of the function y = 3cos(1/3x)?

    -The period of the function y = 3cos(1/3x) is calculated by dividing 2Ο€ by the coefficient of 'x', which is 1/3. So, the period is 2Ο€ / (1/3) = 6Ο€.

  • What is the effect of a negative vertical shift on the graph of a sine or cosine function?

    -A negative vertical shift moves the graph of a sine or cosine function downwards, changing the maximum value from the positive amplitude to the negative of the shift amount.

Outlines
00:00
πŸ“ˆ Basics of Graphing Sine and Cosine Functions

This paragraph introduces the fundamentals of graphing sine and cosine functions, focusing on the basic structures and equations. It starts with the simple equation y = sin(x) and explains the amplitude, period, and how the graph looks. The paragraph then contrasts this with the graph of y = cos(x), highlighting the differences in starting points and the shape of the graphs. The concept of amplitude and period modification is introduced with examples like y = 2sin(x) and y = sin(2x), explaining how they affect the graph.

05:02
πŸ“‰ Negative Functions and Vertical Shifts

The second paragraph delves into the effects of negative signs and vertical shifts on sine and cosine functions. It explains how a negative sign inverts the graph and how vertical shifts affect the midline of the graph. The paragraph provides examples of graphing functions with vertical shifts, such as y = sin(x) + 2, and explains how to calculate the new amplitude and the resulting graph's appearance.

10:02
πŸŒ€ Phase Shifts and Period Adjustments

This paragraph discusses phase shifts and period adjustments in sine and cosine functions. It explains how to calculate the phase shift by setting the function equal to zero and solving for x. The paragraph also covers how to find the period of the function and uses examples like y = 2sin(4x) - 3 to illustrate the process. The explanation includes how to plot key points and extend the graph to cover one full cycle.

15:03
πŸ”„ Domain and Range of Sine and Cosine Functions

The final paragraph addresses the domain and range of sine and cosine functions, emphasizing that these typically extend from negative infinity to infinity unless restricted. It provides an example of graphing y = -2sin(1/3x) - Ο€/2 + 3, detailing how to find the phase shift, calculate the period, and plot the graph. The paragraph concludes by reiterating the importance of understanding amplitude, vertical shift, phase shift, and period when graphing sine and cosine functions.

Mindmap
Keywords
πŸ’‘Amplitude
Amplitude refers to the maximum distance from the center line of a wave to its peak (either positive or negative). In the context of the video, it determines the 'height' of the sine or cosine wave, with a value of 1 for a standard wave. For instance, when the video discusses 'y equals 2 sine', the amplitude is 2, meaning the wave will range from -2 to 2, double the height of a standard sine wave.
πŸ’‘Period
The period of a wave is the length of one complete cycle, from the starting point back to the same point. In the video, the period for a basic sine wave is given as 2Ο€, but it changes with different coefficients in front of the x (e.g., 'sine 2x' has a period of Ο€, and 'cosine 1/2 X' has a period of 4Ο€). The period is calculated using the formula 2Ο€ divided by the coefficient of x.
πŸ’‘Phase Shift
A phase shift is a horizontal translation of the graph of a function. In the video, a phase shift is denoted by a change in the starting point of the wave. It is determined by the value of 'C' in the equation y = a sin(Bx + C) + D. A positive 'C' value shifts the graph to the right, while a negative 'C' shifts it to the left. For example, 'y = 2 sin(4x - 3)' has a phase shift of -3Ο€/2, meaning the graph starts 3Ο€/2 units to the left of the standard 'y = 2 sin 4x' graph.
πŸ’‘Vertical Shift
A vertical shift is a vertical translation of the graph of a function. It is determined by the value of 'D' in the equation y = a sin(Bx + C) + D. A positive 'D' value shifts the graph upwards, and a negative 'D' shifts it downwards. In the video, 'y = sin x + 2' has a vertical shift of 2 units upwards, meaning the entire sine wave is raised by 2 units on the y-axis.
πŸ’‘Sine Function
The sine function is a trigonometric function that models periodic phenomena such as the motion of waves or the oscillation of a pendulum. In the video, the sine function is graphed in various forms, including basic 'y = sin x', transformed with amplitude changes ('y = 2 sin x'), phase shifts ('y = sin(x - 3)'), and vertical shifts ('y = sin x + 2').
πŸ’‘Cosine Function
The cosine function is another trigonometric function that, like the sine function, represents periodic changes. It differs in that cosine starts at the maximum value of the amplitude at the origin, whereas sine starts at the midpoint. The video covers the basic 'y = cos x', as well as modified cosine functions with different amplitudes, periods, phase shifts, and vertical shifts.
πŸ’‘Horizontal Stretch
A horizontal stretch in the context of graphing sine or cosine functions refers to the horizontal scaling of the function's graph. This is achieved by changing the coefficient in front of the x (B in y = a sin(Bx + C) + D). If B is greater than 1, the graph is stretched horizontally, and if B is less than 1, it is compressed. For example, 'y = sin(1/2x)' represents a horizontal stretch because 1/2 is less than 1, making the period twice as long.
πŸ’‘Vertical Stretch
A vertical stretch in the context of graphing sine or cosine functions refers to the vertical scaling of the function's graph. This is represented by the coefficient 'a' in the equation y = a sin(Bx + C) + D. If 'a' is greater than 1, the graph is stretched vertically, and if 'a' is between 0 and 1, it is compressed. For instance, 'y = 2 sin x' represents a vertical stretch because 2 is greater than 1, doubling the amplitude of the sine wave.
πŸ’‘Domain
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For sine and cosine functions, the domain is typically all real numbers, from negative infinity to positive infinity, unless otherwise restricted. The video mentions that the domain for sine and cosine graphs is negative infinity to infinity, indicating that these functions can be graphed over any real number range.
πŸ’‘Range
The range of a function is the set of all possible output values (y-values) that result from the input values within the domain. For sine and cosine functions, the range is typically between -1 and 1 for a standard sine or cosine wave. However, the range can change with different amplitudes and vertical shifts. In the video, the range is discussed in relation to the amplitude and vertical shifts, such as 'y = 3 cos(1/2x) + 5' having a range from -8 to -2.
πŸ’‘Phase
Phase in trigonometric functions refers to the point in the cycle of the wave at which the function begins. It is a measure of the horizontal shift from the standard starting point of a sine or cosine wave, which is typically at the origin (x=0). The phase is determined by the value of 'C' in the equation y = a sin(Bx + C) + D. A phase of 0 indicates no shift, while other values indicate a shift to the left or right.
Highlights

The video focuses on graphing sine and cosine functions, particularly with horizontal phase shifts.

The basic equation y = sin(x) is introduced with an amplitude of 1 and a period of 2Ο€.

Negative sine function (-sin(x)) is explained, showing it goes down first instead of up.

Cosine function (cos(x)) is differentiated from sine, starting at the top and varying between 1 and -1.

The generic equation for sine or cosine wave is given as y = a * sin(Bx + C) + D, where a is amplitude, B affects period, C is horizontal phase shift, and D is vertical shift.

The effect of a vertical stretch in the sine function (e.g., sin(2x)) is explained, doubling the graph in the Y direction.

Horizontal stretching is demonstrated with the example of sin(1/2x), which stretches the graph by a factor of 2.

The concept of vertical shift is illustrated with the function y = sin(x) + 2, showing how to adjust the center line.

An example of combining vertical shift and amplitude is given with y = 3sin(x) + 4, highlighting the process of plotting the midline and range.

The process of graphing a function with a phase shift is detailed, using y = 2sin(4x) - 3 as an example.

The method for calculating the period of a function is explained, using 2Ο€/B for the formula.

The video demonstrates how to find the phase shift by setting the function equal to zero and solving for x.

The range and domain of sine and cosine graphs are discussed, noting that they typically extend from negative infinity to infinity.

A comprehensive example is provided for graphing y = 3cos(1/2x) + Ο€ - 5, including vertical shift, amplitude, phase shift, and period calculation.

The video concludes by summarizing the process for graphing sine and cosine functions with amplitude, vertical shift, phase shift, and how to find all parameters from the equation.

Transcripts

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TrigonometryWave FormsGraphing TechniquesAmplitude AdjustmentPhase ShiftsVertical ShiftPeriod CalculationTrig FunctionsSine & CosineMath EducationTrigonometric Graphs