
Trigonometry 

Graphs of
Trigonometric Functions 
The
Graph of the Function y = asin
(bx + c) 
The function y = asin
x

The function y = sin
bx 
The function y = sin
(x + c)

The function 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
xaxis. 
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
xaxis that is 
f
(x) = sin x => f
(x 
x_{0}) = sin
(x 
c), x_{0} =
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. 










Precalculus contents
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