Describe the effect of parameters in polar curves.
Describe the effect of parameters in polar curves.
Compare polar and Cartesian graphs.
Sketch standard polar graphs.
Identify standard polar graphs.
Write equations for standard polar graphs.
Find intersection points of polar graphs.
Graphing in Polar Coordinates
When we plot points in Cartesian coordinates, we start at the origin and move a distance right or left given by the [latex]x[/latex]-coordinate of the point, then move up or down according to the [latex]y[/latex]-coordinate. When we sketch the graph of an equation or function, we think of drawing the graph from left to right, with the “height” of the graph at each [latex]x[/latex]-value given by the function, as shown in figure (a).
In polar coordinates, however, the dependent variable, [latex]r\text{,}[/latex] gives not a height but a distance from the pole in direction [latex]\theta\text{,}[/latex] as shown in figure (b). When graphing an equation in polar coordinates, we think of sweeping around the pole in the counterclockwise direction, and at each angle [latex]\theta[/latex] the [latex]r[/latex]-value tells us how far the graph is from the pole.
Example 10.17.
Graph the polar equation [latex]r=2\sin \theta\text{.}[/latex]
Solution
We make a table of values, choosing the special values for [latex]\theta\text{.}[/latex] For each value of [latex]\theta\text{,}[/latex] we evaluate [latex]r=2\sin \theta\text{.}[/latex]
[latex]\theta[/latex]
[latex]0[/latex]
[latex]\dfrac{\pi}{6}[/latex]
[latex]\dfrac{\pi}{4}[/latex]
[latex]\dfrac{\pi}{3}[/latex]
[latex]\dfrac{\pi}{2}[/latex]
[latex]\dfrac{2\pi}{3}[/latex]
[latex]\dfrac{3\pi}{4}[/latex]
[latex]\dfrac{5\pi}{6}[/latex]
[latex]\pi[/latex]
[latex]r[/latex]
[latex]0[/latex]
[latex]1[/latex]
[latex]\sqrt{2}[/latex]
[latex]\sqrt{3}[/latex]
[latex]2[/latex]
[latex]\sqrt{3}[/latex]
[latex]\sqrt{2}[/latex]
[latex]1[/latex]
[latex]0[/latex]
First we’ll plot the points in the first quadrant. Observe that as [latex]\theta[/latex] increases from [latex]0[/latex] to [latex]\dfrac{\pi}{2}\text{,}[/latex] [latex]r[/latex] increases from [latex]0[/latex] to [latex]2\text{.}[/latex] Starting at the pole, we connect the points in order of increasing [latex]\theta\text{.}[/latex] Imagine a radial line sweeping around the graph through the first quadrant: as the angle increases, the length of the segment increases so that its tip traces out the graph shown in figure (a). Now continue plotting the points in the table as [latex]\theta[/latex] increases from [latex]\dfrac{\pi}{2}[/latex] to [latex]\pi\text{.}[/latex] In the second quadrant, [latex]r[/latex] decreases as [latex]\theta[/latex] increases, as shown in figure (b). The graph we obtain is, in fact, a circle, which we will prove algebraically shortly. However, we have not yet plotted points for [latex]\theta[/latex] between [latex]0[/latex] and [latex]2\pi\text{.}[/latex] Because [latex]\sin \theta[/latex] is negative in the third and fourth quadrants, all the [latex]r[/latex]-values for these angles ar