Match each graph to its corresponding equation. Justify your...

Match each graph to its corresponding equation. Justify your answers (two equations have no match). Graph cannot copy. a. $r=\frac{10}{5+5 \sin \theta}$ b. $r=\frac{8}{4-2 \cos \theta}$ c. $r=\frac{54}{3-2 \sin \theta}$ d. $r=\frac{4.2}{3+2 \sin \theta}$ e. $r=\frac{12}{4+6 \sin \theta}$ f. $r=\frac{6}{2-2 \cos \theta}$ g. $r=\frac{4}{2+3 \sin \theta}$ h. $r=\frac{9}{3+6 \cos \theta}$

Match each graph to its corresponding equation. Justify your answers (two equations have no match). Graph cannot copy.
a. $r=\frac{10}{5+5 \sin \theta}$
b. $r=\frac{8}{4-2 \cos \theta}$
c. $r=\frac{54}{3-2 \sin \theta}$
d. $r=\frac{4.2}{3+2 \sin \theta}$
e. $r=\frac{12}{4+6 \sin \theta}$
f. $r=\frac{6}{2-2 \cos \theta}$
g. $r=\frac{4}{2+3 \sin \theta}$
h. $r=\frac{9}{3+6 \cos \theta}$

Key Concepts

Polar Coordinates

A system for representing points in the plane using a distance from a reference point and an angle from a reference direction. This approach is especially useful when dealing with curves that exhibit radial symmetry or have a focus at the origin.

Conic Sections in Polar Form

Conic sections can be expressed in polar coordinates with equations of the form r = ed/(1 ± e sin?) or r = ed/(1 ± e cos?). These forms simplify the expression of ellipses, parabolas, and hyperbolas, particularly when one focus is located at the pole.

Eccentricity

Eccentricity (e) is a parameter that determines the type and shape of a conic section. In polar equations, the value of e indicates whether a curve is an ellipse (e < 1), a parabola (e = 1), or a hyperbola (e > 1), and it influences the curvature of the graph.

Orientation of Conic Sections

The use of sine or cosine in the denominator of a polar conic equation signals the direction in which the conic is oriented. Specifically, a sine term generally leads to a vertical orientation, while a cosine term often results in a horizontal orientation of the graph.

Graph-Equation Matching

Matching a graph to its corresponding polar equation involves analyzing the orientation, shape, vertex location, and directrix properties as indicated by the structure of the equation, and then finding these features in the plotted graph.

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