Sine Function


General Sine Functions

The trigonometric sine function
f(x) = a sin(b x + c) + d

,its amplitude, period and phase shift are explored interactively using an html 5 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:
f(x) = a sin(b x + c) + d
in blue and
g(x) = a sin(b x) + 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 .

Interactive Tutorial Using Html 5 applet


sin(x)
a =
b =
c =
d =
>

How do the 4 coefficients a, b, c and d affect the graph of f(x)?

Amplitude

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|.

Answer

Period

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.

Answer

Phase Shift

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?

Answer

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?

Answer

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

Answer

Vertical Shift

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

Answer

More References and Links on Sine Functions

Explore interactively the Derivatives of Sine (sin x) Functions
Match Sine Functions to Graphs. Excellent activity where graphs and functions are matched.
Explore interactively the sum of a sine and a cosine functions
Sum of Sine and Cosine Functions
Examples with detailed solutions and explanations on sine function problems.
Tutorial on Sine Functions (1)- Problems
Tutorial on the relationship between the amplitude, the vertical shift and the maximum and minimum of the sine function
Tutorial on Sine Functions (2)- Problems
Trigonometric Functions
Step by step graphing of sine functions
Graph of Sine, a*sin(bx+c), Function
Explore interactively the relationship between the graph of sine function and the coordinates of a point on the unit circle
Unit Circle and Trigonometric Functions sin(x), cos(x), tan(x)
The Six Trigonometric Functions Calculator.