Trigonometry
Graphs of Trigonometric Functions
y = asin (bx + c)
y = asin x
The function y = sin bx
y = sin (x + c)
y = asin (bx + c)
The graph of the function  y = a sin (b x + c)
The trigonometric functions of this form have wide application in physics, electricity and engineering where are used in analyzing and modeling behavior of different situations in which things follow a repeating pattern.
Therefore, we need to know how the parameters a, b and c affect the form of the source sine function              y = sin x.
The function y = a sin x
The graph of the function asin x is obtained by multiplying each function value sin x by the constant a.
That means, the ordinates of each point of the function sin x should be a times,
- increased if  a > 0
- decreased if  0 < a < 1
- while if  a < 1 each point of the graph should at the same time be flipped around x-axis.
The parameter a is called amplitude.
The function y = sin bx
The parameter b indicates the number of times function repeats itself within the period of 2p. Since there are b periods of the given function in 2p, then the length of its period is  P = 2p/b.
The principal period P should satisfy identity for the periodic functions, thus
 For example, least or principal period of the function sin 2x, What means, its graph
repeats twice within the period of 2p.
 While the function has a half of its period within the interval of 2p, as its principal
 period as can be seen in the figure below.
The function y = sin (x + c)
The parameter c represents the value of translation of the sin x function in the direction of the x-axis that is
f (x) = sin x  =>    f (x - x0) = sin (x - c),   x0 = c.
For example, the graph of the function  y = sin (x - p/6) is obtained by translating the sin x function horizontally to the right by x = p/6, as shows the graph below.
The function y = asin (bx + c)
The graph of the sin x function repeat once while its argument passes through all the values of an interval,        [x, x + 2p] of the length 2p.
Therefore, the function y = a sin (bx + c) will repeat once while its argument (bx + c) passes through all the values from 0 to 2p, that is
0 < bx + c < 2p
from where  bx + c > 0   =>     x > - c/b     and    bx + c < 2p   =>    x < - c/b + 2p/b.
That means, the given function will start its period at  x = - c/b   and end at the point x = - c/b + 2p/b.
It follows that the least or principal period of the function y = a sin (bx + c)P = 2p/b.
 For example the function will repeat once in the interval that is
 Thus, given function will repeat once in each interval of the length p, or P = p, while
the initial point of the given interval is at x = - p/6,  as is shown in the figure below.
Pre-calculus contents G