Sine Function

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The trigonometric sine function

f(x) = a*sin(bx+c)+d

,its amplitude, period and phase shift are explored interactively using an applet. The investigation is carried out by changing the parameters a, b, c and d. To deeply understand the effects of each parameter on the graph of the function, we change one parameter at the time at the start. Then later we may change more than one parameter.
Exploration and understanding of the phase shift is done by comparing the shift between the graphs of the two functions:
in blue and

g(x) = a*sin(bx) + d

in red as shown in the figure below.

sine function with and without phase shift

You may also want to consider another tutorial on the trigonometric unit circle .

Once you finish the present tutorial, you may want to work through a self test on trigonometric graphs .

There are two types of applets that may be used to explore general sine functions

1) Interactive Tutorial Using Sine Function HTML5 applet

2) Interactive Tutorial Using Java Applet

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How do the 4 coefficients a,b,c and d affect the graph of f(x)?
  1. use the scrollbar to set a=1,b=1,c=0 and d=0. Write down f(x) and take note of the amplitude, period and phase shift of f(x)? Now change a , how does it affect the graph? The amplitude is defined as |a|.


  2. set a=1,c=0,d=0 and change b. Find the period from the graph and compare it to 2pi/|b|. How does b affect the graph of f(x)? The period is the horizontal distance (along the x-axis) between two points: one is the starting point of a cycle and the second is the end point of the same cycle.


    Phase Shift
  3. set a=1,b=1,d=0 and change c starting from zero going slowly to positive larger values. Take note of the shift, is it left or right?


  4. set a=1,b=1,d=0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right?


  5. repeat the above for b=2,3 and 4, measure the shift and compare it to -c/b (the phase shift).


    Vertical Shift
  6. set a,b and c to non zero values and change d. What is the direction of the shift of the graph?


More references and links on sine functions.