Here is a sample of a past piece of coursework, studying the effects on sine and cosine graphs. This GCSE coursework is quite old and lacks mathematical reasoning. Please do not take the information you read here as difinitive.
The effect of varying the coefficients on a sine or cosine graph will be investigated. The graph below shows sin x over 720 degrees from -360 degrees to 360 degrees. This shows the familiar shape of the sine graph.
The x-axis will represent degrees and y value of sin or cos . This is the standard manner of setting out graphs and will save confusion. Red will indicate sine and green will represent cosine.
I have noticed that both sin and cos graphs have a similar appearance. The cos graph is 90 degrees right along the x-axis. It has been moved +90 degrees along the x-axis. There maybe a rule to translate the one graph to equal the same values as the other.
I used a spreadsheet to show the graphs. Initially I had problems as Excel was recognising sin x and cos x using radians and not degrees. I will be using degrees so a formula was needed to convert the radians.
I found a formula for translating radians into degrees using the help facility on Excel. There was a formula using Pi which goes as follows:
=SIN(A2*PI()/180) for Excel or Sin(x × Pi ÷ 80).
Transformation Description Translation The graph/object has been moved from one set of co-ordinates to another. This movement is described using vectors [ ], with x at the top and y at the bottom. Reflection This is when there is a mirror image of the graph/object. The line of symmetry is given as y =, for example y = 2x. Rotation When a graph/object has been turned around it has been rotated. The angle of rotation must be given as well as the centre of rotation, for example (2, 0) 90 degrees clockwise. Enlargement If a graph/object is made larger, smaller, stretched or squeezed it is enlargement. Enlargement can be written as fractions or as a ratio, for example; ¼ or 1:4 .
These diagrams show characteristics of coefficients a, b, c and d where y = a sin (b x + c) + d. All coefficients are positive and above 1.
- - - - - previous graph -------- current graph -------- sine -------- cosine vertical spread of graph increases
oscillations increase graph transposed left graph transposed up
Here are my final conclusions on y = a sin (b x + c) + d, all points shown below are from my research and investigating.
When the value of the coefficient is positive:
a increases the vertical spread of the graph by its value
b increases the oscillations per 360o by its value
c moves the graph left along the x-axis by its value
d moves the graph up the x-axis by its value
When the value of the coefficient is less than 1:
a decreases the vertical spread of the graph by its value
b decreases the oscillations per 360o by its value
c moves the graph left along the x-axis by < 1
d moves the graph up the x-axis by < 1
When the value of the coefficient is negative:
a inverts the vertical spread of the graph by its value
b increases the oscillations per 360o and inverts the graph by its value
c moves the graph right along the x-axis by its value
d moves the graph down the x-axis by its value
Translating sin x and cos x:
sin x = cos (x-90)
cos x = sin (x+90)
The peak of the graph a + d
The trough of the graph d - a
Oscillations per 360o b
Interesting facts associated sine:
Normally cosine and sine are used to determine waves.
In music, if the frequency of two notes were sin x and sin 2x they would be octaves.
In music, if the frequency of two notes were sin x and sin 1.5x they would be fifths.
The formula for frequency is: speed/wavelength and measured in hertz (Hz).