Topic 3.14 Notes – Polar Function Graphs

What a Polar Function Graph Is

A polar function has the form $r=f(\theta)$. Here, $\theta$ tells you the direction and $r$ tells you the signed distance along that direction.

• The polar axis is the positive $x$-axis.
• Positive angles go counterclockwise. Negative angles go clockwise.
• If $r>0$, plot on the terminal ray of $\theta$.
• If $r<0$, plot the point the other way, opposite that ray.
• If $r=0$, the point is the origin.

The actual distance from the origin is $|r|$, not $r$.

The two panels below show that sign difference for the same angle $\theta=\frac{\pi}{6}$. In panel (b), $(2,\frac{\pi}{6})$ lies on the ray for $\frac{\pi}{6}$. In panel (a), $(-2,\frac{\pi}{6})$ lands the same distance away in the opposite direction.

Equivalent polar points matter a lot:

$$(r,\theta)=(r,\theta+2\pi k), \qquad (r,\theta)=(-r,\theta+(2k+1)\pi)$$

So one physical point can have many polar names.

A quick check in rectangular coordinates is

$$x=r\cos\theta,\qquad y=r\sin\theta$$

but the main job here is graphing as polar, not converting.

One exam trap shows up constantly. The rectangular graph of $(\theta,f(\theta))$ is not the polar graph.

How to Construct the Graph

When you build a polar graph from a formula or table, the curve is traced as $\theta$ increases.

1. Identify the domain of $\theta$. Only graph that interval.
2. Pick useful angles, usually unit-circle angles.
3. Compute $r=f(\theta)$.
4. Plot each polar point $(r,\theta)$, handling negative $r$ correctly.
5. Connect points in increasing $\theta$ order.
6. Sketch smoothly if the function is continuous.
7. Mark endpoints if the domain is restricted.

Angles worth prioritizing in a table:

• where $r=0$ so the graph hits the origin
• where $r$ changes sign
• where $r$ is largest or smallest
• where trig values are exact
• where the domain starts or ends

If you’re given a table, each row is an input-output pair $(\theta,r)$, but the plotted point is the polar point $(r,\theta)$, not the rectangular point $(\theta,r)$.

Restricted Domains and Endpoints

A restricted domain means you keep only the part traced by those allowed $\theta$-values.

The endpoints come from the polar pairs $(f(a),a)$ and $(f(b),b)$, not just from the physical location. That matters because the same point can happen at different angles. The origin is the biggest source of confusion since $r=0$ works for any angle.

If an endpoint angle is included, that point is on the graph. If the endpoint angle is excluded, leave it off.

When a problem shows part of a curve and asks for the domain, you need both:

• the endpoint angles
• the path followed as $\theta$ increases

Patterns to Notice While Sketching

Some features tell you what kind of curve you’re getting.

• Origin crossings happen when $f(\theta)=0$.
• Sign changes in $r$ often flip the trace to the opposite side.
• Loops happen when the curve leaves a point, closes a piece, and returns.
• Self-intersections happen when different inputs give equivalent polar points.

Periodicity helps, but be careful.

• The period of $f$ can help find the full graph.
• The graph might close before one full period of angle values.
• Some curves retrace over $0\le\theta\le2\pi$.
• Some need longer intervals.

Examples you should recognize include a circle from a constant radius, a spiral from $r$ growing with $\theta$, and a trig form that creates a looped shape.

• $r=c$ gives a circle centered at the origin.
• $r=\theta/2$ gives a spiral.
• Trig forms can create circles, indented curves, inner loops, and petal-like shapes.

Symmetry and Common Mistakes

Useful symmetry tests:

• $f(-\theta)=f(\theta)$ gives symmetry about the polar axis
• $f(\pi-\theta)=f(\theta)$ gives symmetry about the vertical axis
• $f(\theta+\pi)=f(\theta)$ gives origin symmetry
• $f(\theta+\pi)=-f(\theta)$ often means the graph retraces itself

Use symmetry to save work, but confirm it from the formula.

Common mistakes:

• treating $(r,\theta)$ like rectangular coordinates
• forgetting negative $r$ goes opposite the ray
• connecting nearest-looking points instead of increasing $\theta$
• graphing the whole curve when the domain is restricted
• assuming $0\le\theta\le2\pi$ is always enough
• confusing signed radius $r$ with distance $|r|$