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Periodic Function
y=sin(x)
y=cos(x)
Periodic functions are functions that repeat
after a period of time, such as sine and cosine functions. The
sine function passes through the origin in its original format,
splitting the wave half way between the max and min of the wave.
The cosine is has its maximum portion of the wave on the y axis
where x=o.
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Reflection (x)
The reflection is done by making the entire
function negative, this makes the function reflect over the x
axis. When reflected across the x axis each y value becomes the
opposite value. The function is now written as
y=(-)sin(x) or y=(-)cos(x).
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Period
The period is the length of time it takes
before the function will repeat again. For y=sin(x) or y=cos(x)
(the parent functions) the period will be 2 pie (360 degrees).
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Amplitude
The amplitude is the distance from half
of the wave to the top of the wave (half way between the minimum
point and maximum point is half the wave). For the parent function
y=sin(x) the amplitude is 1.
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Horizontal
To create a horizontal stretch the number
would need to be divided from the x value of the parent function.
To stretch the function one would write y=sin(x/a) or y=cos(x/a).
To compress the function the opposite application would be done,
multiplication to the x. This would be written as y=sin(ax) or
y=cos(ax).
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Vertical
To do the vertical stretch the multiplication
(compress) and division (stretch) need to be applied outside the
parenthesis. These functions would look like y=sin(x)/a or y=cos(x)/a
for the stretch. For the compression they would be y=sin(x)a or
y=cos(x)a.
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Horizontal
To apply a horizontal stretch the number
must be added to the x value inside the parenthesis. y=sin(x+a)
or y=cos(x+a). In both cases the shift is to the left, the positive
moves it left as subtraction will shift the function right. An
example of a function shifted right would be y=sin(x-a) or y=cos(x-a).
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Vertical
To apply a vertical shift to a periodic
function you add or subtract the amount at the end of the equation
(to the y value). The number added will be the amount it is raised
on the y axis. The number when subtracted will be the amount it
is shifted down the y axis.
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