Algebra 2: Sine and Cosine Functions

Questions and Answers

Which of the following have a frequency factor of b = 1? (Select all that apply)

A sine function whose graph shows 2 cycles from -4π radians to 0 (correct)

A cosine function whose graph shows 1 cycle from 3π radians to 5π radians (correct)

A cosine function with no phase shift whose x-coefficient is -1

A sine function whose period is 2π radians (correct)

When would two sine functions of the form y = sin(x - h) that have different values for h have the same graph?

If the h-values differ by multiples of 2π.

Which of the following is the graph of y=cos(2(x+π))?

Graph D

Graph A

Graph B (correct)

Graph C

Which term gives the horizontal length of one cycle of a periodic function?

Period Signup and view all the answers

What is the period of y = sin(3x)?

2π/3 Signup and view all the answers

Which term gives the number of cycles of a periodic function that occur in one horizontal unit?

Frequency Signup and view all the answers

Which transformations are needed to change the parent cosine function to y=0.35cos(8(x-π/4))?

Vertical compression of 0.35, horizontal compression to a period of π/4, phase shift of π/4 units to the right Signup and view all the answers

The graph of y=sin(x-3π/2) is the graph of the y = sin(x) shifted in which direction?

3π/2 units to the right Signup and view all the answers

What is the equation of the graph below?

y=cos(x+π) Signup and view all the answers

Which of the following could be the equation of the function below?

y=-2cos(2(x+π))-1 Signup and view all the answers

What is the phase shift of a periodic function?

A horizontal translation of the function Signup and view all the answers

What is the general equation of a sine function with an amplitude of 2, a period of π, and a horizontal shift of π units?

y=2sin(2(x-π)) Signup and view all the answers

Study Notes

Frequency and Period

A sine function with a period of 2π has a frequency factor ( b = 1 ).
The period of a function refers to the horizontal length of one complete cycle.
The frequency indicates the number of cycles occurring in one horizontal unit.

Sine Function Characteristics

Different sine functions ( y = \sin(x - h) ) can have the same graph if the ( h ) values differ by multiples of ( 2π ).
The period of ( y = \sin(3x) ) is ( \frac{2π}{3} ).

Transformations of Trigonometric Functions

To transform the parent cosine function to ( y = 0.35\cos(8(x - \frac{π}{4})) ):
- Apply a vertical compression by a factor of 0.35.
- Apply a horizontal compression resulting in a period of ( \frac{π}{4} ).
- Implement a phase shift ( \frac{π}{4} ) units to the right.

Graphs and Phase Shifts

The graph of ( y = \sin(x - \frac{3π}{2}) ) represents a shift of ( \frac{3π}{2} ) units to the right.
The equation of its cosine counterpart can be expressed as ( d: y = \cos(x + π) ).
An example function is given by ( y = -2\cos(2(x + π)) - 1 ).

Key Terms

Period: Defines the horizontal length of one cycle in periodic functions.
Phase Shift: Represents the horizontal translation of the function, altering its starting point along the x-axis.

Description

Test your knowledge on the changes in period and phase shift of sine and cosine functions with these flashcards. Understand the frequency factors and describe the behavior of the graphs through various scenarios. Perfect for reinforcing concepts in Algebra 2.