# Variations of the Sine Function

by
### Patricia Johnson

Consider the equation **y = a sin(bx + c) **for different values of
**a**

The value of **a **determines the amplitude of the wave. For example,
when a = 2 the sine function oscillates between +2 and -2; when a=5 the
function oscillates between +5 and -5. When **a **is negative we get
a symmetry reflection of the function in the y=0 axis. When a =0 the function
is then represented by the line y=0 i.e. there is no displacement.

Now let us consider the function when only **b** varies:

This graph shows that by varying **b **only** **the period of the
wave changes. If we consider those graphs when **b **is positive, we
notice that as **b** increases the wavelength of the wave decreases but
the frequency increases. When negative values of **b** are compared with
the corresponding positive values we notice a phase shift in the waves.

We will now examine when only **c** varies:

We notice that as **c** increases for positive values, the waves experiences
a phase shift to the left. When **c** increases for negative values,
the wave shifts to the right.

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