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

Sine and Cosine Components in Polar Equations

The appearance of either sine or cosine in the denominator of a polar conic section's equation affects its orientation. Equations featuring sine typically indicate that the conic is oriented such that its directrix is horizontal, while those with cosine suggest a vertical orientation. This difference is crucial when comparing the algebraic form with the graph's layout.

Scaling and Shifting in Polar Equations

The constant multipliers in the numerator and the additive constants in the denominator of these polar equations dictate the scaling and positional shift of the conic section relative to the pole. Evaluating these parameters helps in determining the size, location, and orientation of the conic, which is key for matching each graph with its corresponding equation.

Polar Conic Sections

These equations represent conic sections expressed in polar coordinates, where the focus is located at the pole. Their standard form typically resembles r = ed/(1 ± e sin?) or r = ed/(1 ± e cos?), with e representing the eccentricity and d relating to the position of the directrix. Recognizing the structure of these equations is essential for understanding and matching the corresponding graphs.

Eccentricity

Eccentricity indicates the shape of a conic section. In a polar equation, if the eccentricity (e) is less than 1, the conic is an ellipse; if e equals 1, the conic is a parabola; and if e is greater than 1, the conic is a hyperbola. Identifying the eccentricity from the equation helps in determining the conic section’s form and, consequently, matching it with its graph.

Transcript

Please give Ace some feedback