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Period and Frequency

Horizontal distance traveled before y values repeat; number of complete waves in 2pi.

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Vertical and Horizontal Translations, Amplitude, Period, and Frequency

Feel free to modify and personalize this study guide by clicking "Customize".

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Keep in mind that the x-axis will most likely be in Radians, if not it will be in Degrees.

 vertical translation is a shift in a graph up or down along the "y" axis, generated by adding a constant to the original function.

Hint: Vertical means upright, or going up and down. The "y" axis is upright also

License: CC BY-NC 3.0

This is what a vertical translation would look like on a graph but in a function you would have to add or subtract by a number depending if you want to shift the graph up or down.

\begin{align*}y=\sin(x)\end{align*}

\begin{align*}y=\sin(x)+2\end{align*}

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Horizontal Shifts

This is the same as a vertical translation only that it shifts the graphs left or right

\begin{align*}y=\sin(x-\frac{\pi}{2})\end{align*}

Caution: When shifting the graph, keep in mind the subtraction or addition sign, these depict whether the graph shifts to the left or to the right. If there is a negative sign then the graph shift to the left but if there is a positive sign then then the graph shifts to the left!

License: CC BY-NC 3.0
Tip:You can transform a cosine graph into a sine graph by shifting it to the right by \begin{align*}\frac{\pi}{2}\end{align*}.


Amplitude:

The amplitude of a wave is a measure of the wave's height.

License: CC BY-NC 3.0

Tip: For the image above, the center line is at y=0 and the amplitude affects the height above and below that line by two!

How can the range be affected by changing the amplitude?

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Period and Frequency

The period of a wave is the horizontal distance traveled before the 'y' values begin to repeat.

License: CC BY-NC 3.0

Sine and cosine will always will always have a period of 2\begin{align*}\pi\end{align*} but will not be 2\begin{align*}\pi\end{align*}when the function is changed

TIP: just remember that the period is when the the values will repeat again.

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The frequency of a wave is number of complete waves every \begin{align*}2 \pi\end{align*} units.

Since the period and frequency is inversley related, what happes to the period as the frequency increases? Click here for guidence.

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\begin{align*}y=D \pm A \cos(B(x \pm C))\end{align*} or \begin{align*}y=D \pm A \sin(B(x \pm C))\end{align*} , where \begin{align*}A\end{align*} is the amplitude\begin{align*}B\end{align*} is the frequency, \begin{align*}C\end{align*} is the horizontal translation, and \begin{align*}D\end{align*} is the vertical translation.

Tip: Always start with the vertical translation and then horizontal translation so that it makes the graphing much easier.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0

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