Use a graphing utility to graph the polar equation. Describe the resulting graph. $r = 4 \cos ( 7 \theta - 2 )$
A) rose curve with 7 petals, rotated 2 radians
B) rose curve with 7 petals, rotated 4 radians
C) lemniscate rotated 4 radians
D) limaçon rotated 4 radians
E) cardioid rotated 2 radians

Question

Quizplus · Accepted Answer

The equation $r = 4 \cos ( 7 \theta - 2 )$ represents a rose curve because of the cosine function with a multiple of $\theta$ inside the argument. The number 7 indicates there will be 7 petals if $7$ is odd, which it is. The phase shift (rotation) is determined by the equation inside the cosine function, which is $7\theta - 2$. To find the actual rotation, you solve for $\theta$ when the inside equals 0, which gives $\theta = \frac{2}{7}$ radians. However, the choice mentioning rotation directly is incorrect in its specifics; the correct aspect is the presence of 7 petals, which aligns with the general form of a rose curve equation $r = a\cos(k\theta)$ or $r = a\sin(k\theta)$, where $k$ determines the number of petals. The rotation detail provided in the options is misleading; the key aspect is the structure of the curve (7 petals) dictated by the coefficient of $\theta$ inside the cosine function.

Use a Graphing Utility to Graph the Polar Equation r=4cos⁡(7θ−2)r = 4 \cos ( 7 \theta - 2 )r=4cos(7θ−2)

Use a Graphing Utility to Graph the Polar Equation $r = 4 \cos ( 7 \theta - 2 )$