Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.r² = 4 sin θ

Accepted Answer

Recognize that the given equation is in polar form: $$r^{2} = 4$$ \sin 	heta$$. Our goal is to understand the shape of this curve by analyzing and possibly converting it to Cartesian coordinates or by studying its behavior in polar coordinates. Recall the relationships between polar and Cartesian coordinates: x = r$$ \cos 	heta$$, y = r$$ \sin 	heta$$, and r$$^{2}$$ = x$$^{2}$$ + y$$^{2}$$. These will help us rewrite the equation in a more familiar form if needed. Substitute r$$^{2}$$ = x$$^{2}$$ + y$$^{2}$$ and $$\sin 	heta = \frac{y}{r}$$ into the equation: r$$^{2}$$ = 4 \sin 	heta$$ becomes $$x^{2}$$ + $$y^{2} = 4$$ \cdot \frac{y}{r}$$. Multiply both sides by r$ to eliminate the denominator, remembering that r$$ = \sqrt{$$x^{2}$$ + $$y^{2}$$}$$. After multiplying, you get (x$$^{2}$$ + y$$^{2}$$) \cdot r = 4y. $$Substitute $$r = \sqrt{x$$^{2}$$ + y$$^{2}$$}$$ back in to get $$(x^{2}$$ + $$y^{2}$$) \sqrt{$$x^{2}$$ + $$y^{2}$$} = 4y$$. This is a Cartesian form that can help identify the curve's shape. Analyze the resulting equation or use the original polar form to plot points for various values of $$	heta$$ between 0$ and 2$$\pi$$. Note the symmetry and key points (like where r=0$ or r$ is maximum) to sketch the graph. Finally, use a graphing utility to verify your sketch.

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.

r² = 4 sin θ

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Key Concepts

Polar Coordinates and Polar Equations

Graphing Polar Curves

Using Graphing Utilities for Polar Graphs