
Trigonometry 

Graphs of
Trigonometric Functions 
The
Graph of the Function y = acos
(bx + c) 
The
graphs of the functions,
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
= 
2, b
= 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 xaxis 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 x_{0}
= p/4
and the ending point at x_{0}
= 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
= 1, b
= 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 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,














Precalculus contents
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