Trace polar curves instead of treating radius as height

Try

Trace r=4cos θ. At θ=0, the point is (4,0). At θ=π/2 the radius is zero, placing the point at the origin. At θ=π the radius is −4, which places the point back at (4,0), not at (−4,0). A polar table must convert the signed radius and angle together.

Predict

Predict whether the curve is a circle centred at the origin. Its farthest positive horizontal point is 4, while it passes through the origin, suggesting a shifted circle instead. Predict whether letting θ run from 0 to 2π traces new points throughout the entire interval.

Experiment

Compute and plot several exact points.

θ	r=4cos θ	x	y
0	4	4	0
π/4	2√2	2	2
π/2	0	0	0
3π/4	−2√2	2	−2
π	−4	4	0

Connect the points in parameter order. One full traversal occurs from 0 to π; the next π radians retrace the same circle. The curve lies to the right of the origin even when its signed radius becomes negative.

Understand

Multiply r=4cos θ by r and use r²=x²+y² and r cos θ=x. This gives x²+y²=4x, or (x−2)²+y²=4: a circle with centre (2,0) and radius 2. Multiplication by r can make an equation automatically true at r=0, so conversion should be checked there. In this example the origin already occurs at θ=π/2, so no new Cartesian point has been introduced.

For r=2+2cos θ, radius is always nonnegative and becomes zero at θ=π. The graph has a cusp at the origin. For r=3cos2θ, radius changes sign and creates a four-petal rose. A few samples illustrate its structure but do not alone establish every branch; use symmetry and intervals between zeros to organize a full trace.

Radial change is not vertical change. For r=2+cos θ from 0 to π, radius changes from 3 to 1, so average radial change is −2/π per radian. Both Cartesian endpoint heights are zero because sin0=sinπ=0. The direction has changed along with radius.

Likewise, increasing signed radius need not increase distance from the origin: moving from r=−3 to r=−1 decreases |r|. Graph claims must identify whether they concern signed radius, actual distance, x-coordinate or y-coordinate.

Apply

In practice, retain signed radii and identify repeated traversal. For a converted circle, state centre and radius and check the origin. For the average-change task, preserve the angular denominator and distinguish a radial rate from a Cartesian component rate.

Continue

Complex polar form uses the same radius-and-angle representation for a different operation: multiplication. Products add angles and multiply radii, connecting algebra directly to rotation and scaling.