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

Eccentricity

Eccentricity (e) in a polar conic equation determines the type and shape of the conic section. For example, when 0 < e < 1 the curve is an ellipse, when e = 1 it is a parabola, and when e > 1 it is a hyperbola. Understanding the value of eccentricity helps in predicting the curvature and the extent to which the conic deviates from being circular.

Orientation and Directrix in Polar Forms

The orientation of the conic section in the polar coordinate system is governed by whether the trigonometric function in the denominator is sine or cosine. A sine function typically aligns the conic’s symmetry with the vertical axis, while a cosine function aligns it with the horizontal axis. The plus or minus sign affects the location of the directrix relative to the pole, thereby influencing the graph’s orientation.

Polar Equations

Polar equations express curves using a radial coordinate r and an angular coordinate ?, differing from Cartesian equations. They are especially useful for representing curves that have circular or rotational symmetry and are fundamental in understanding the geometric behavior of the curves plotted in a polar plane.

Conic Sections in Polar Coordinates

Conic sections—such as ellipses, parabolas, and hyperbolas—can be represented in polar coordinates with equations of the form r = ed/(1 ± e sin?) or r = ed/(1 ± e cos?). These forms highlight how the geometry of the conic relates to the focus (usually at the origin) and a directrix, encapsulating the curve’s shape in terms of eccentricity and directrix distance.

Graph Matching and Analysis

Matching graphs to their polar equations involves analyzing parameters such as the coefficient in the numerator (which relates to the directrix distance) and the terms in the denominator (which indicate orientation and eccentricity). By comparing these parameters, one can determine which curve corresponds to a particular algebraic representation, even if the graphs are not explicitly provided.

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