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| Trigonometry |
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Graphs of
Trigonometric Functions |
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The
Graph of the Function y = a
sin
(bx + c) |
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The function y = asin
x
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The function y = sin
bx |
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The function y = sin
(x + c)
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The function y = a
sin
(bx
+ c) |
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| The
graph of the function y = a
sin
(bx + 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. |
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Therefore, we need to know how the parameters a,
b
and c
affect the form of the source sine function
y =
sin x. |
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| The function y = a
sin
x |
| The graph of the function
a sin 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 negative,
each point of the graph should at the same time be flipped around
x-axis. |
| The parameter
a
is called amplitude. |
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The function y = sin
b
x |
| 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.
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| The principal period
P
should satisfy identity for the periodic functions, thus |
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| For example, least or principal period of the function
sin 2x, |
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What means, its graph |
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repeats twice within the period of
2p. |
| While the function |
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has a half of its period within the
interval of 2p, as its principal |
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| period |
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as can be seen in
the figure below. |
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The function y = sin
(x + c) |
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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.
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The function y = a
sin
(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. |
| Hence,
the least or principal period of the function
y = a
sin (bx + c),
P = 2p/b.
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| For example the function |
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will repeat once in the interval |
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that is |
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Thus, given function will repeat once in each interval of the length
p, or
P = p, while
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| the initial point of the given interval is at
x = -
p/6,
as is shown in the figure below. |
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| Trigonometry
contents B |
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