Question

match each polar equation (a - d) with its associated polar graph.
a. ( r = 1 + 2cos\theta )
b. ( r = 1 + 2sin\theta )
c. ( r = 1 - 2cos\theta )
d. ( r = 1 - 2sin\theta )

• a solution write - up is not required for this problem.

Explanation:

To solve this, we analyze the general form of polar equations for conic sections (specifically cardioids here). The standard form for a cardioid with a cosine term ($r = a \pm b\cos\theta$) opens horizontally, and with a sine term ($r = a \pm b\sin\theta$) opens vertically.

• For $r = 1 + 2\cos\theta$ (Option A): The positive cosine term means it opens to the right (since $\cos\theta$ is maximum at $\theta = 0$). The coefficient of $\cos\theta$ is 2, and $a = 1$, $b = 2$ (since $b > a$, it's a cardioid? Wait, no—actually, for $r = a + b\cos\theta$, if $a = b$, it's a cardioid, but here $a = 1$, $b = 2$, so it's a limaçon with an inner loop? Wait, no, the problem is about matching to a graph (though the graph isn't shown here, but typically, let's recall:

Wait, maybe the original problem had graphs (not shown here), but assuming the standard matching:

• $r = 1 + 2\cos\theta$ (A) has a horizontal axis (cosine) and opens right (positive sign).
• $r = 1 + 2\sin\theta$ (B) has vertical axis (sine) and opens up (positive sign).
• $r = 1 - 2\cos\theta$ (C) has horizontal axis, opens left (negative sign on cosine).
• $r = 1 - 2\sin\theta$ (D) has vertical axis, opens down (negative sign on sine).

But since the user's problem is to match, and assuming the typical graphs, but since the graphs are not provided, maybe there was a typo or missing info. However, if we assume the question is to identify the correct equation based on a graph (e.g., a cardioid opening right would be $r = 1 + 2\cos\theta$ (A), opening up $r = 1 + 2\sin\theta$ (B), etc.). But since the original problem's graphs are missing, but if we proceed with the options, let's clarify:

Wait, the problem says "Match each polar equation (A - D) with its associated polar graph." Since the graphs are not shown, but maybe in the original context, the correct match depends on the graph's orientation. For example, if the graph opens to the right, it's $r = 1 + 2\cos\theta$ (A); if it opens up, $r = 1 + 2\sin\theta$ (B); left: $r = 1 - 2\cos\theta$ (C); down: $r = 1 - 2\sin\theta$ (D).

But since the user's image only shows the options, maybe the intended answer is based on standard matching. Let's assume the graph (not shown) corresponds to one of these. For example, if the graph is a cardioid opening to the right, the answer is A. If it opens up, B, etc. But since the problem is incomplete (no graph), but assuming a typical problem, let's proceed.

Wait, maybe the user made a mistake, but assuming the question is to select the correct equation (e.g., if the graph is a cardioid opening right, the answer is A: $r = 1 + 2\cos\theta$).

Answer:

A. $r = 1 + 2\cos\theta$ (assuming the graph matches this equation, e.g., opening right horizontally)