*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.*

** **A ** 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*

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.

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#### 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!*

**Tip:**You can transform a cosine graph into a sine graph by shifting it to the right by .

#### Amplitude:

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

**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.

Sine and cosine will always will always have a period of 2 but will not be 2when 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 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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or , where is the amplitude, is the frequency, is the horizontal translation, and is the vertical translation.

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