Match the polar equations with the graphs labeled I-IX. Give...

Match the polar equations with the graphs labeled I-IX. Give reasons for your choices. (a) $r=\cos 3 \theta$ (b) $r=\ln \theta, \quad 1 \leqslant \theta \leqslant 6 \pi$ (c) $r=\cos (\theta / 2)$ (d) $r=\cos (\theta / 3)$ (e) $r=\sec (\theta / 3)$ (f) $r=\sec \theta$ (g) $r=\theta^{2}, \quad 0 \leqslant \theta \leqslant 8 \pi$ (h) $r=2+\cos 3 \theta$ (i) $r=2+\cos (3 \theta / 2)$

Match the polar equations with the graphs labeled I-IX. Give reasons for your choices.
(a) $r=\cos 3 \theta$
(b) $r=\ln \theta, \quad 1 \leqslant \theta \leqslant 6 \pi$
(c) $r=\cos (\theta / 2)$
(d) $r=\cos (\theta / 3)$
(e) $r=\sec (\theta / 3)$
(f) $r=\sec \theta$
(g) $r=\theta^{2}, \quad 0 \leqslant \theta \leqslant 8 \pi$
(h) $r=2+\cos 3 \theta$
(i) $r=2+\cos (3 \theta / 2)$

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and angle, making them ideal for describing curves with radial symmetry or angular periodicity. Understanding this system is essential when interpreting and sketching polar graphs.

Graphing Polar Equations

Graphing in polar coordinates involves plotting points derived from the equation r as a function of ?. Key techniques include analyzing symmetry, periodicity, and domain restrictions, which are crucial when matching given equations to their corresponding graphs.

Rose Curves

Rose curves are a family of polar graphs given by equations such as r = cos(n?) or r = sin(n?), where n determines the number of petals (n petals if n is odd, and 2n petals if n is even). Recognizing these curves is important for identifying graphs that exhibit rotational petal patterns.

Spiral Curves

Spiral curves, like those defined by r = ?^2 or r = ln ?, feature a continuously increasing or decreasing radius as the angle increases. This non-periodic expansion or contraction distinguishes spirals from other polar graphs and helps in their identification.

Limaçons

Limaçons are represented by equations of the form r = a + b cos? or r = a + b sin?. Their shapes can vary from dimpled to having an inner loop or being convex, depending on the values of a and b. This variability in shape is key to recognizing and matching different polar graphs.

Secant and Reciprocal Trigonometric Functions

Equations involving reciprocal trigonometric functions, such as r = sec?, lead to graphs with asymptotic behavior or discontinuities. Understanding how these functions transform the graph—often by introducing vertical asymptotes or gaps—is important when identifying these curves.

Angle Transformations

Modifying the argument of trigonometric functions (for example, using multiples or fractions of ?) affects the period and symmetry of the resulting graph. These transformations are essential for predicting features such as the number of petals in rose curves or the repeating sections in other polar graphs.

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