Which of the following equations would graph a lemniscate?

Question

(1 point)
Responses

lower r squared equals 9 sine 2 theta
Image with alt text: lower r squared equals 9 sine 2 theta

r = 2 sin 9 θ
r = 2 sin 9 θ

lower r squared equals 9 minus sine 2 theta
Image with alt text: lower r squared equals 9 minus sine 2 theta

r = 9 + 2 sin θ

Answer 1

To determine which of the given equations graphs a lemniscate, we can analyze the nature of each equation.

1. \( r^2 = 9 \sin(2\theta) \): This is the equation of a lemniscate. The presence of the \(\sin(2\theta)\) factor suggests that it will create a figure-eight shape.

2. \( r = 2 \sin(9\theta) \): This represents a rose curve with multiple petals but does not graph a lemniscate.

3. \( r^2 = 9 - \sin(2\theta) \): While this could represent a different type of curve depending on its coefficients, it does not resemble the classic lemniscate form.

4. \( r = 9 + 2 \sin(\theta) \): This describes a different type of polar graph, specifically a shifted sinusoidal graph, not a lemniscate.

The equation that graphs a lemniscate is:

\( r^2 = 9 \sin(2\theta) \).