Using the Polar Tracker Tool to Investigate How Polar Roses are Drawn

In this exploration, you will gain some intuition on how polar curves are drawn. Additionally, you will learn that the same polar curve can be drawn in many different ways.

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Plot r1(t) = sin(2t) on 0t2.
Click the Polar Tracker tool on the tool palette in the Graph window. It is the sixth tool from the left.
Move the arrow cursor over the radial line source (a large black circle) on the right end of the Polar axis.
Press and hold down the mouse button. Drag the radial line in a counterclockwise direction.

As you drag the mouse, a radial line will follow. The radial line represents the subtended angle measured in radians from the polar axis. As you drag the radial line, a black dot moves along the plotted polar curve, representing the value of the polar function for the current value of t. When r is positive, the dot, radial line, and polar curve all intersect. However, when the value of r is negative, the dot appears on the curve in the quadrant diagonally opposite the quadrant containing the radial line. As you drag the polar tracker, the coordinates of the point on the polar curve are displayed in the bottom two cells of the Domain & Range window.
Drag the radial line counterclockwise in a complete circle.
Note that the petals of the rose are drawn in the following order: quadrant 1, quadrant 4, quadrant 3, and quadrant 2.

Plot r2(t) = sin(-2t) on 0t2.
Drag the radial line counterclockwise in a complete circle.
Note that the petals of the rose are drawn in the following order: quadrant 3, quadrant 2, quadrant 1, and quadrant 4.

Find two other polar equations that draw this same four petal rose but draws the petals in a different order on 0

Select New Function from the Work menu and enter your polar equation.
Plot your polar equation on 0t2.
Use the Polar Tracker tool to see if your polar equation gives a different ordering for the petals.

The two hidden functions shown in the Work window contain the expressions for two possible equations with different petal orderings. Plot them and use the polar tracker to determine the ordering of the petals. Are they the same as yours?

To see the expression of a hidden function,

First select the hidden function in the Work window, and second,
Select Show Hidden Function from the Options menu.

Work window

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