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Vertical and Horizontal Translations, Amplitude, Period, and Frequency
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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.




Horizontal Shifts

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


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 \frac{\pi}{2}.


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?


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\pi but will not be 2\piwhen the function is changed

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


The frequency of a wave is number of complete waves every 2 \pi units.

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


y=D \pm A \cos(B(x \pm C)) or y=D \pm A \sin(B(x \pm C)) , where A is the amplitudeB is the frequency, C is the horizontal translation, and D 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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