Determine which equation belongs to the graph of the limacon curve below. [-5,5] by [-5,5]

Your question seems to refer to a specific graph of a limacon, but unfortunately, since this is a text-based interface, I don't have access to the graph. However, I can explain how you can match an equation to a limacon graph.

Answer: To match an equation to a limacon curve, you first need to know the standard form of a limacon's equation, which is given in polar coordinates as:

r = a + b*cos(θ) or r = a + b*sin(θ)

Where: - r is the radius (or distance from the origin to a point on the curve) - θ is the angle in polar coordinates - a and b are constants that determine the shape and size of the limacon

The nature of the limacon—whether it has a loop, is a dimpled limacon, or is a cardioid—depends on the relationship between a and b: - If |a| > |b|, the limacon will not have a loop. - If |a| = |b|, the limacon is a cardioid (a special case of the limacon). - If |a| < |b|, the limacon will have a loop.

Additionally, the location of the loop or the dimple (in the case of dimpled limacons) depends on whether the cosine or sine function is used.

To determine the equation from a graph, follow these logical steps: 1. Identify if the limacon has a loop, a dimple, or is a cardioid. 2. Look at the symmetry to decide whether cosine or sine should be used. 3. Determine the point(s) where the curve crosses the polar axis (the horizontal axis in polar coordinates); this will give you information about the constants a and b. 4. If the curve reaches its maximum distance from the origin at θ = 0 or π, use cosine. 5. If the curve reaches its maximum distance from the origin at θ = π/2 or 3π/2, use sine. 6. Using the identified properties and points on the curve, you should be able to derive values for a and b and construct the equation.