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# Transformations of Polar Graphs

## Alteration of graph based on changing constants and/or function of a polar equation.

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Polar Coordinates and Graphs

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### Vocabulary

##### Complete the chart.
 Word Definition Polar Coordinates ________________________________________________________ ______________ a graph of two heart shaped loops reflected across the "x" axis ______________ a graph with a sinusoidal curve looping around the origin Transformation ________________________________________________________

### Polar Coordinates

How many radians are in a circle? _____________

How do you convert from degrees to radians? ____________________________

In your own words, explain how to graph polar points on a graph: _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Name the coordinates of the points on this graph:



A: ____________

B: ____________

C: ____________

Graph the following on a calculator:

1.  r=1+3sinθ\begin{align*}r = 1 + 3 sin \theta\end{align*}
2. r=1+2cosθ\begin{align*}r = 1 + 2 cos \theta\end{align*}
3. (1.5,2π3)\begin{align*}\left(1.5, \frac{2\pi}{3}\right)\end{align*}

Give three alternate sets of coordinates for the given point within the range 360θ360\begin{align*}-360^\circ \leq\theta\leq 360^\circ\end{align*} .

1. (3,120)\begin{align*}(3, 120^\circ )\end{align*}
2. (1,240)\begin{align*}(1, 240^\circ )\end{align*}
3. (4,345)\begin{align*}(4, 345^\circ )\end{align*}

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#### Distance Between Two Polar Coordinates

What is the Law of Cosines? ______________________________________

How do we have to transform the Law of Cosines to make it a distance formula for polar coordinates? _____________________________________________________

Therefore, what is the Polar Distance Formula? ______________________________________

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Find the distance between each set of points:

1. (3,260)\begin{align*}(-3, 260^\circ )\end{align*} and (2,90)\begin{align*}(2, 90^\circ )\end{align*}
2. (4,45)\begin{align*}(4, -45^\circ )\end{align*} and (6,150)\begin{align*}(6, 150^\circ )\end{align*}
3. (5,60)\begin{align*}(-5, -60^\circ )\end{align*} and (1,250)\begin{align*}(1, 250^\circ )\end{align*}
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#### Transformations

Give an equation for a line on a polar graph: _______________________________

Give an equation for a circle on a polar graph: _______________________________

Graph the following polar equations on the same polar grid and compare the graphs.

rr=5+5sinθ=5(1+sinθ)r=55sinθr=5(1sinθ)\begin{align*}r & = 5 + 5 \sin \theta && r = 5 - 5 \sin \theta \\ r & = 5(1 + \sin \theta) && r = 5(1 - \sin \theta) \end{align*}

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What are the graphs of r=a±bsinθ\begin{align*}r = a \pm b \sin \theta\end{align*} or r=a±bcosθ\begin{align*}r = a \pm b \cos \theta\end{align*} called? _________________________

Graph each function using your calculator and sketch on your paper.

1. r=23cos(θ)\begin{align*}r=2 -3\cos(\theta )\end{align*}
2. r=1+2sin(θ)\begin{align*}r=1+2\sin(\theta )\end{align*}
3. r=2+5cos(θ)\begin{align*}r=-2+5\cos(\theta )\end{align*}
4. How do sine and cosine graphs differ?
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