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

Polar coordinates represent points in the plane by a distance from the origin (r) and an angle (?) from a reference direction. This coordinate system is often used when dealing with curves that have rotational symmetry or radially dependent properties, such as conic sections with one focus at the pole.

Conic Sections in Polar Form

Many conic sections—ellipses, parabolas, and hyperbolas—can be expressed in polar coordinates by equations of the form r = ed/(1 + e cos?) or r = ed/(1 + e sin?). In these expressions, the constants are arranged so that one focus of the conic is at the pole, with the directrix and eccentricity determining its shape and orientation.

Eccentricity

Eccentricity (e) is a key parameter that determines the type of conic section: if e < 1 the conic is an ellipse, if e = 1 it is a parabola, and if e > 1 it is a hyperbola. In the polar equations, the value of e is obtained after rewriting the equation into a standard form, and it describes how 'stretched' the conic is.

Axis of Symmetry in Polar Conics

The trigonometric function (sine or cosine) in the denominator of a polar conic equation indicates the direction of the conic’s axis of symmetry. A sine function suggests that the conic is symmetric about a vertical line (aligned with the polar axis rotated by 90°), whereas a cosine function implies symmetry about a horizontal line, aiding in the identification and matching of graphs to equations.

Graph Analysis and Matching

Matching an algebraic polar equation to its corresponding graph involves analyzing the equation parameters to identify characteristics such as the conic type, eccentricity, directrix, and orientation. By comparing these features abstractly, one can justify which equation produces a graph with a specific shape and position in the polar coordinate system.

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