Algebra II Unit 11: Graphs and Amplitude

Questions and Answers

What are sound waves modeled using?

Sine function (correct)

Cosine function

Linear function

Quadratic function

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?

|A| Signup and view all the answers

What are asymptotes?

Lines that a graph approaches but never touches. Signup and view all the answers

What is the domain in relation to a function?

The set of first coordinates in a relation; the set of independent variable (x) values. Signup and view all the answers

What does frequency refer to?

The number of times the pattern in the graph repeats within $2π$. Signup and view all the answers

What is the period of a function?

The interval of the domain in which one cycle of the graph is completed. Signup and view all the answers

What is a periodic function?

A function that repeats a pattern over its domain. Signup and view all the answers

What does the range of a function represent?

The set of second coordinates in a relation; the set of dependent variable (y) values. Signup and view all the answers

What does the function f(x) = sinx represent?

Both A and B Signup and view all the answers

What is the sine value range?

-1 ≤ y ≤ 1 Signup and view all the answers

What happens when restricting the domain?

You restrict the graph to a specific section of the sine curve. Signup and view all the answers

What is the cosine function equation?

y = cosx Signup and view all the answers

What is the amplitude of the function y = 2cosx?

2 Signup and view all the answers

How is the function y = sin2x evaluated?

By first doubling the value of x, and then taking the sine. Signup and view all the answers

What is Hertz?

The base unit for frequency. Signup and view all the answers

What is the formula to find the period from frequency?

P = 2π/f Signup and view all the answers

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!