Tiffany N. KeysŐ Assignment 1:

All
About Sines

The
graphs of all sine functions are related to the graph of y = sinx

This
graph has the following characteristics:

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The
**domain** of the function is all real
numbers.

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The**
range** of the function is -1 __<__ y __<__
1 or [-1, 1]

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The
function is **periodic**, meaning that its graph has a repeating pattern that
goes on indefinitely.

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The
shortest repeating portion of the pattern is called a **cycle**
and the horizontal length of each cycle is the **period**.

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The
amplitude of the function is half the
vertical distance between its minimum and its maximum value.

*INVESTIGATION*: Examine graphs of y = *a* sin (*b*x +*c*) for
different values of a, b, and c.

First,
letŐs begin with substituting in values for *a*:

Y = sin(x) y = 2 sin(x) y = 3 sin(x)

**Y =
-sin(x) **** y = -2 sin(x) y = -3 sin(x)**

OBSERVATIONS:

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The
graph of each function passes through the origin, (0,0).

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Each
graph is continuous along the x-axis and appears to repeat itself once it
reaches a certain point on the x-axis.

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The
domain of all the graphs is all real numbers, however they reach have different
ranges.

They are as follows:

Y = sin(x), [-1, 1] **Y = -sin(x), [-1,1]**

y = 2 sin(x) , [-2, 2] **y = -2 sin(x), [-2,2]**

y = 3
sin(x), [-3, 3] **y =
-3 sin(x), [-3,3]**

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The
amplitude of the graph is affected when the value of a
is changed. When a > 1, the slope of the
graph to appears to be more steep.
When a < 1, the magnitude of the
amplitude is the absolute value of a and the
graph is reflected across the x-axis.

Next,
letŐs observe what happens when different values for *b*
are substituted in:

Y = sin(x) y = sin(2x) y = sin(3x)

**Y =
sin(-x) **** y = sin(-2x) y = sin(-3x)**

OBSERVATIONS:

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Like
the graphs of each function above, these also pass through the origin, (0,0).

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They
are also continuous along the x-axis and appear to repeat themselves once they
reach a certain point on the x-axis.

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The
domain of all the graphs is all real numbers and they all have the same range
of [-1, 1].

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When
b is
changed in the function, the period is effected. When **b**
> 1, the graph squeezes and the period becomes shorter.** **When b <
1, the periods stayed the same and, like when a was assigned a negative value,
the graph is reflected across the x-axis.

Lastly,
letŐs notice the difference in the graph when values for *c* are substituted in:

Y = sin(x
+ 1) y = sin(x + 2) y = sin(x + 3)

**Y =
sin(x - 1) **** y = sin(x - 2) y = sin(x - 3)**

OBSERVATIONS:

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Unlike
the graphs above, these graphs do not pass through the origin.

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They
are, however, continuous along the x-axis and repeat themselves once they reach
a certain point on the x-axis.

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The
domain of all the graphs is all real numbers and they all have the same range
of [-1, 1].

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The
part of the graphs that are effected when the c value
is changed in the function is the phase shift, which is the graphŐs movement
along the horizontal axis. When **c** > 1, the graph moves to the right along
the x-axis and** w**hen
c < 1, the phase shift is toward the left
along the x-axis.