Question

match each graph with its equation
a. $r = \cos(5\theta)$
b. $r = \cos(7\theta)$
c. $r = \cos(6\theta)$
d. $r = \sin(6\theta)$
e. $r = \cos(4\theta)$

Explanation:

Step1: Recall the formula for the number of petals in a rose - curve

For a polar equation of the form \(r = a\cos(n\theta)\) or \(r=a\sin(n\theta)\), if \(n\) is odd, the number of petals is \(n\). If \(n\) is even, the number of petals is \(2n\).

Step2: Analyze the first graph

The first graph has \(12\) petals. Since \(n = 6\) (even) in \(r=\cos(6\theta)\) and \(r = \sin(6\theta)\), and for \(n\) even, the number of petals is \(2n\). For \(r=\cos(6\theta)\), the graph is symmetric about the \(x -\)axis. For \(r=\sin(6\theta)\), the graph is symmetric about the \(y -\)axis. The first graph is symmetric about the \(x -\)axis, so it matches \(r=\cos(6\theta)\) (option c).

Step3: Analyze the second graph

The second graph has \(12\) petals and is symmetric about the \(y -\)axis. Using the symmetry property ( \(r=\sin(n\theta)\) is symmetric about the \(y -\)axis for polar equations of the form \(r = a\sin(n\theta)\)), it matches \(r=\sin(6\theta)\) (option d).

Step4: Analyze the third graph

The third graph has \(8\) petals. Since \(n = 4\) (even) in \(r=\cos(4\theta)\), and for \(n\) even, the number of petals is \(2n\). So \(2n=8\) when \(n = 4\). Also, \(r=\cos(4\theta)\) is symmetric about the \(x -\)axis. So it matches \(r=\cos(4\theta)\) (option e).

Answer:

The first graph matches c. \(r=\cos(6\theta)\), the second graph matches d. \(r = \sin(6\theta)\), and the third graph matches e. \(r=\cos(4\theta)\)