match - polar - equations - and - graphs\nr = 6 - cos(θ)\nr = 5 - 11/9θ\nr² = 49 cos(2θ)\nr = 10 cos(3θ)

Explanation:

Step1: Recall polar - curve characteristics

The general forms of some polar curves help in identification. For a polar equation of the form $r = a\pm b\cos\theta$ or $r=a\pm b\sin\theta$, if $a = b$, it is a cardioid; if $a>b$, it is a limacon without an inner - loop; if $a < b$, it is a limacon with an inner - loop. The equation $r = a\cos(n\theta)$ or $r=a\sin(n\theta)$ is a rose curve. If $n$ is odd, the rose has $n$ petals, and if $n$ is even, the rose has $2n$ petals. The equation $r^{2}=a^{2}\cos(2\theta)$ or $r^{2}=a^{2}\sin(2\theta)$ is a lemniscate. The equation $r = a + b\theta$ is a spiral.

Step2: Analyze $r = 6-\cos(\theta)$

This is of the form $r=a - b\cos\theta$ with $a = 6$ and $b = 1$. Since $a>b$, it is a limacon without an inner - loop.

Step3: Analyze $r = 5-\frac{11}{9}\theta$

This is of the form $r=a + b\theta$, so it is a spiral.

Step4: Analyze $r^{2}=49\cos(2\theta)$

This is of the form $r^{2}=a^{2}\cos(2\theta)$ with $a = 7$, so it is a lemniscate.

Step5: Analyze $r = 10\cos(3\theta)$

This is of the form $r=a\cos(n\theta)$ with $n = 3$ (odd), so it is a rose curve with 3 petals.

Answer:

• $r = 6-\cos(\theta)$: Limacon without inner - loop (matches with the appropriate graph among the given ones which is a non - looped limacon)
• $r = 5-\frac{11}{9}\theta$: Spiral (matches with the appropriate graph among the given ones which is a spiral)
• $r^{2}=49\cos(2\theta)$: Lemniscate (matches with the appropriate graph among the given ones which is a lemniscate)
• $r = 10\cos(3\theta)$: Rose curve with 3 petals (matches with the appropriate graph among the given ones which is a 3 - petaled rose curve)