The graph of the equation r² = −16sin2θ is a Limaçon with one key point at the origin (0, 0), where −16 affects the size and direction of the loop, and the number 2 affects the period of oscillation.
The equation r² = −16sin2θ represents a type of graph known as a Limaçon. The significance of the −16 in the equation is related to the shape and size of the Limaçon, indicating that it is an inner loop Limaçon with a loop radius of 4 units, since the coefficient of the sine function determines the size of the loop, and the negative sign indicates that the loop is on the left of the pole for positive θ values.
The number 2 in the sine function suggests that the Limaçon will complete a full oscillation as θ goes from 0 to π, not the usual 2π for a regular sine function because of the double angle. There are generally four key points associated with a Limaçon, but because of the specific form of this equation, where the value after the equal sign is constant, the graph will be a circle with a radius of −4 in polar coordinates, and there would only be one key point, which is the origin (0, 0).