Trigonometry
Graphs of Trigonometric Functions
y = acos (bx + c)
y = asin (bx + c) and  y = acos (bx + c), examples
The graph of the function  y = a cos (b x + c)
The parameters a, b and c have the same influence on the graph of the function cos x as to the function
sin x, since we already know that the cosine function is translated sine function and vice versa.
Thus, for example the function
 repeats once in the interval that is within
Therefore, the function repeats itself at every interval of the length 4p or the period P = 4p and the initial
point of the given interval at x = p/2.
At the same time it means that the graph of the given function can be obtained translating the function
Example:   Examine the properties, draw the graph and analyze behavior of the function
y = - 2sin (2x/3 - p/6).
Solution:  Comparing with  y = a sin (bx + c) it follows that,  a = - 2b = 2/3  and  c = - p/6.
The influences of the given parameters to the shape and the position of the graph in a coordinate system we
can examine and analyze on the following way,
- since a < 0 the graph of the given function, relating to the graph of the source function y = sin x, is
flipped around the x-axis and bounded by lines  y = - 2 and  y = 2.
 - The least or principal period of the function therefore, the function
repeats once within the interval 3p.
 - Horizontal translation of the graph what means that the given period will
have its initial point at x0 = p/4 and the ending point at x0 = p/4 + 3p.
That is, the function will repeat itself once within the interval  x Î [p/4, p/4 + 3p].
- Zeros of the function y = a sin (bx + c) we calculate from
- The abscissas of extremes ( maximums and minimums) of the given function (or y = a sin (bx + c)) we
 calculate from
According to the properties drown is the graph of the given function.
 Behavior of the function within the principal period,
Example:   Examine the properties, draw the graph and analyze behavior of the function
y = cos (3x/2 + p/2).
Solution:  Comparing with  y = a cos (bx + c) it follows that,  a = 1b = 3/2  and  c = p/2.
The given parameters determine the properties,
- since a = 1, the graph of the function is bounded between  y = -1 and  y = 1,
 - the principal period

 - the translation
thus the given interval  x Î [- p/3, p].
- the zeros of the function  y = a cos (bx + c)  we calculate using the formula
- the abscissas of the extremes of the function  y = a cos (bx + c)  we calculate using the formula
 The behavior of the function within the principal period,
Pre-calculus contents G