*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.

\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!*

**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.

**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\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.*