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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.  $r = 1 + 3 sin \theta$
2. $r = 1 + 2 cos \theta$
3. $\left(1.5, \frac{2\pi}{3}\right)$

Give three alternate sets of coordinates for the given point within the range $-360^\circ \leq\theta\leq 360^\circ$ .

1. $(3, 120^\circ )$
2. $(1, 240^\circ )$
3. $(4, 345^\circ )$

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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. $(-3, 260^\circ )$ and $(2, 90^\circ )$
2. $(4, -45^\circ )$ and $(6, 150^\circ )$
3. $(-5, -60^\circ )$ and $(1, 250^\circ )$
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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.

$r & = 5 + 5 \sin \theta && r = 5 - 5 \sin \theta \\r & = 5(1 + \sin \theta) && r = 5(1 - \sin \theta)$

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What are the graphs of $r = a \pm b \sin \theta$ or $r = a \pm b \cos \theta$ called? _________________________

1. $r=2 -3\cos(\theta )$
2. $r=1+2\sin(\theta )$
3. $r=-2+5\cos(\theta )$
4. How do sine and cosine graphs differ?
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