The graph $x^2 + y^2 = 9$ can be considered the set of all solutions to the Pythagorean theorem $a^2 + b^2 = c^2,$ where $c$ is set at 3. Several points on the graph (with approximations) are shown above. Also note:

$\begin{aligned} (-1.10)^2 + (2.79)^2 &\approx 9 \\ (2.46)^2 + (1.72)^2 &\approx 9 \\ (-2.12)^2 + (-2.12)^2 &\approx 9 \end{aligned}$

What does the entire graph of $x^2 + y^2 = 9$ look like?

In a traditional $xy$-coordinate system from the previous problem, the first value $x$ indicates horizontal distance and the second value $y$ indicates vertical distance. In a **polar coordinate system** using $(r, \theta),$ the value $r$ gives "distance forward from origin when facing angle $\theta.$" Some example points are shown above.

One curious thing is that $r$ can be negative. Which point is $(-3, 135^\circ) ?$

Other than the same point being identified with a positive $r$ (facing forward) or a negative $r$ (facing backward), it's possible for the same point to be identified in an infinite number of ways via the measure of angle.

The point below can be identified as $(2, 180^\circ) .$ Which of the choices also identifies the same point?

**only negative values**. It can still be used to help to answer the question "Which pair of coordinates refer to the same point?"

Because of possible negative values, polar graphs are often restricted to particular values of $r$ and $\theta .$

If we restrict $\theta$ to positive values only, what kind of graph will $r= \theta$ be?

The visualization below lets you see the effect of $b$ in the spiral graph $r = b \theta .$

What will happen to the graph if $b$ is allowed to be negative?

Polar coordinates give the flexibility to include whole new classes of graphs, like the flower with overlapping petals above. Many other marvels await: start now with an intuitive look at the six main functions of trigonometry.