### GRAPH SKETCHINGANALYSIS OF SINE GRAPHS

in the following activities, post your responses as a comment.

Activity 1:

Define the following terms, where possible identify the context when it is applied.
A. Amplitude
B. Wavelength
C. Frequency
D. Phase
E. Characteristics of the Wave

Activity 2:
Study the following graphs with reference to y = sinx and (1) identify the transformation (s) of the graphs eg. enlargement, shift in x-direction. (2) State whether the new graph is similar to y = sinx. (3) suggest possible the 'transformed' equation. Note that x is in radian.

original
y = sinx

transformation 1

transformation 2

transformation 3

transformation 4

1. Amplitude - The maximum extent of a vibration or oscillation, measured from the rest position.
Wavelength - The shortest distance between any two points in phase.
Frequency - The number of complete waves produced per second.
Phase - Points along a wave are in phase if they have the same direction, speed and displacement from the rest position.
Characteristics of the Wave - The four basic characteristics of a wave are its amplitude, wavelength, speed and phase constant.

- Aziel, Balram and Jun Jie

2. Amplitude: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position

Wavelength: The distance measured in the direction of propagation of a wave between two successive points in the wave, crest to crest or trough to trough

Frequency: Number of waves produced in one second

Phase: a stage in a process (general)

Characteristics of waves: Longitudinal or transverse, period and frequency always the same when passing through same medium,

3. y = A sin(ax+b) + c

A — Changes the amplitude to A.
a — Changes the frequency. The period is multiplied by a.
(eg. in the length of one wavelength of y = sin(x), there are exactly 2 wavelengths of y = sin(2x).)
b — Moves the graph back by b.
c — Changes the y-intercept.

By Aziel, JunJie and Balram.

1. examples:

Change of Amplitude: y = Asinx
Amplitude is changed to A
eg. y = 2sinx
Amplitude is changed to 2

Moving the graph horizontally: y = sin(x+b)
Graph is moved to the left by b
eg. y = sin(x+1)
Graph is moved one unit to the left

Moving the graph vertically: y = sinx + c
Graph's y-intercept is changed to c
eg. y = sinx + 1
Graph's y-intercept is now 1

4. A. Amplitude-the height of the wave from rest point
B. Wavelength-The distance between two crest or trough.
C. Frequency-How many waves in a second
D. Phase-the distance between the first zero-crossing and the point in space defined as the origin. In phase means two waves oscillation is in sync, out of phase means it isn't
E. Characteristics of the Wave-What defines the wave

Transformation 1: 2sin(x)
Transformation 2: sin(2x)
Transformation 3: sin(x+C)
Transformation 4: sin(x)+2

1. A. Amplitude-the height of the wave from rest point
B. Wavelength-The distance between two crest or trough.
C. Frequency-How many waves in a second
D. Phase-the distance between the first zero-crossing and the point in space defined as the origin. In phase means two waves oscillation is in sync, out of phase means it isn't
E. Characteristics of the Wave-What defines the wave

Transformation 1: 2sin(x) -Amplitude increase and graph is no longer similar to first graph.

Transformation 2: sin(2x)-Frequency change and graph is similar

Transformation 3: sin(x+C) or cos(x)-Change of X intercept, graph is simliar

Transformation 4: sin(x)+2-Change of Y intercept, graph is similar

5. A. Amplitude is the angle between the real axis of an Argand diagram and a vector representing a complex number.

B. Wavelength is the distance between successive crests of a wave, esp. points in a sound wave or electromagnetic wave.

C. Frequency is the number of waves occurring in 1 second.

D. Phase is the relationship in time between the successive states or cycles of an oscillating or repeating system and either a fixed reference point or the states or cycles of another system with which it may or may not be in synchrony.

E. Characteristics of the wave
It is described using the wavelength, Phase, period, frequency and amplitude of the wave

Transformation 1: y=2sin(x)
Transformation 2: y=sin(2x)
Transformation 4: y=sin(x)+2

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7. A. Amplitude: Maximum displacement from
B. Wavelength: 2 consecutive points in the same phase
C. Frequency: Number of waves produced per second
D. Phase: The relationship in time between the successive states or cycles of an oscillating or repeating system
E. Characteristics of the Wave: Consists of a trough and a crest.

Activity 2:
Original - sin(x)
transformation 1 - 3sin(x) (Increases the amplitude)
transformation 2 - sin(2x) (Increases the frequency)
transformation 3 - sin(x+c) (moves the graph forward)
transformation 4 - sin(x) +2 (moves the graph up)

Done by : Kai Chek, Hao En and Jee Hoon

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