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 (denoted by e) is a key parameter that determines the type and shape of a conic section. When 0 < e < 1, the conic is an ellipse; when e = 1, it is a parabola; and when e > 1, it is a hyperbola. In the polar equations provided, comparing the coefficient linked to the trigonometric function (cos? or sin?) to the constants allows one to determine the eccentricity of each conic, which in turn influences how the graph appears.

Graphical Analysis of Conic Sections

This concept involves interpreting the shape, orientation, and other features of the graph of a conic section based on its polar equation. By analyzing characteristics such as opening direction, width, and location relative to the pole, one can assess which equation fits which graph. This analytical approach is essential when multiple equations are presented and only some match the available graphs.

Orientation and Symmetry in Polar Graphs

The presence of either cosine or sine in the polar equations influences the orientation of the conic section. Equations with cosine shifts tend to be symmetric about the polar axis, whereas those with sine shifts are symmetric about the line ? = ?/2. Recognizing these symmetries helps in matching the algebraic equation to the observed graph, as the orientation indicates which trigonometric function should be present.

Polar Equations of Conic Sections

This concept involves expressing conic sections (i.e., ellipses, parabolas, and hyperbolas) in polar form. A general polar equation of a conic can be written as r = ed/(1 + e*cos?) or r = ed/(1 + e*sin?), where e represents the eccentricity and d is the distance from the directrix. Understanding this form is essential for matching graphs with their corresponding equations, as it establishes the connection between the algebraic expression and the geometric shape of the curve.

Directrix and Focus

In the context of polar conic sections, the focus is typically placed at the pole (origin), while the directrix is a line perpendicular to the polar axis. The given equations encode the distance to the directrix in the numerator and the effect of the eccentricity in the denominator. This relationship is fundamental to understanding how the graph of the conic is positioned relative to the pole and its orientation in the plane.

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