Questions and Answers
What are sound waves modeled using?
What is amplitude?
The maximum displacement from the horizontal line of symmetry for the sine and cosine functions.
How does the amplitude affect sound?
The smaller the amplitude, the softer the sound.
What is the amplitude for y = Asinx?
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What are asymptotes?
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What is the domain in relation to a function?
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What does frequency refer to?
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What is the period of a function?
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What is a periodic function?
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What does the range of a function represent?
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What does the function f(x) = sinx represent?
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What is the sine value range?
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What happens when restricting the domain?
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What is the cosine function equation?
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What is the amplitude of the function y = 2cosx?
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How is the function y = sin2x evaluated?
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What is Hertz?
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What is the formula to find the period from frequency?
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Study Notes
Sound Waves
- Modeled using the sine function, which provides insights into various sound attributes.
Amplitude
- Maximum displacement from the horizontal line of symmetry in sine and cosine functions.
- Determines loudness: Smaller amplitude equals softer sound; larger amplitude results in louder sound.
- Amplitudes for sine functions:
- For y = sin(x): 1
- For y = -sin(x): 1
- For y = 2sin(x): 2
Functions and Amplitude
- In functions y = Asin(x) and y = Acos(x), the amplitude is |A|.
Asymptotes
- Lines approached by a graph but never touched.
Domain
- Set of first coordinates (x-values) in a relation, representing independent variable values.
Frequency
- Number of graph pattern repetitions within 2π for sine, cosine, and reciprocal functions; within π for tangent and cotangent functions.
- In physics, refers to waves that pass a point per unit of time.
Period of a Function
- Displacement in x where a function's graph begins to repeat, defining one complete cycle.
Periodic Function
- A function that repeats a pattern over its domain.
Range
- Set of second coordinates (y-values) in a relation, representing dependent variable values.
The Function f(x) = sin(x)
- Domain includes all x values (angle measures).
- Range is the sine values of those angles, commonly using radian measure.
The Graph of sin(x)
- Key x-values include: 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π.
- Corresponding sine values are: 0, 0.5, 0.71, 1, 0.71, 0.5, 0, and 0.
Trigonometric Functions
- Type of periodic function where all coterminal angles have identical trig function values.
Sine Function
- Defined for all angle measures; domain is all real numbers; range lies between -1 and 1.
- Sine values are constrained to -1 ≤ y ≤ 1.
Domain Restriction
- Restricts the graph to a specific segment of the sine curve.
Cosine Function
- Defined as y = cos(x).
- Domain is all real numbers; range is -1 ≤ y ≤ 1, repeating every 2π intervals.
Graphing Sin(x) and Cos(x)
- Only need to illustrate from 0 to 2π due to periodicity.
Key Concepts of Trigonometric Functions
- Domain for both sine and cosine is all real numbers.
- Range for both is -1 ≤ y ≤ 1.
- Both are classified as periodic functions.
Reflections and Stretches
- y = -x² reflects the graph of y = x² over the x-axis.
- y = 2x² represents a stretch of the graph of y = x².
Example: Graphing f(x) = 2cos(x)
- Amplitude changes to 2 while maintaining the same shape; range becomes -2 ≤ y ≤ 2.
Function Evaluation: y = sin(2x)
- Calculated by firstly doubling x's value and then applying the sine function.
- Example: For x = π/2, sin(2x) = sin(π) = 0
Frequency Function
- Represents how often a pattern repeats within 2π.
- Function of the form y = sin(Bx), where B indicates frequency; thus, P = 2π/f.
Hertz
- Base unit of frequency measuring one cycle per second.
Calculating Period and Frequency
- For y = sin(2x), frequency (f) is 2, leading to period (P) = π.
- The period for y = sin(2x) is π, indicating its frequency is 2.
Final Review Points
- Amplitude for both y = Acos(x) and y = Asin(x) is |A|.
- Coefficient B influences frequency, with the period defined as P = 2π/f.
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Description
Explore the fundamental concepts of sound waves and amplitude through this quiz. Understand how amplitude affects the sound generated by sine and cosine functions. Perfect for reinforcing your knowledge of graphs in Algebra II!