Question

identify and graph the polar equation.
r = 7^{\theta}
what shape does the polar equation form when graphed?
a. spiral
b. rose
c. limacon
d. cardioid
which graph represents the equation?
a.
b.
c.
d.

Explanation:

Step1: Simplify the polar equation

Given \(r = 7^{\theta}\). This is in the form of a polar equation \(r = a^{\theta}\) (\(a> 0,a
eq1\)), which represents a spiral.

Step2: Recall the characteristics of different polar - curve shapes

• A rose curve has the form \(r = a\cos(n\theta)\) or \(r=a\sin(n\theta)\).
• A limacon has the form \(r=a + b\cos\theta\) or \(r=a + b\sin\theta\).
• A cardioid is a special case of a limacon with \(a = b\), i.e., \(r=a(1+\cos\theta)\) or \(r=a(1 + \sin\theta)\).

Since the given equation \(r = 7^{\theta}\) does not match the forms of rose (\(r = a\cos(n\theta)\) or \(r=a\sin(n\theta)\)), limacon (\(r=a + b\cos\theta\) or \(r=a + b\sin\theta\)), or cardioid (\(r=a(1+\cos\theta)\) or \(r=a(1+\sin\theta)\)) curves, but matches the form of a spiral (\(r=a^{\theta}\)).

Answer:

A. Spiral