Graphing r = θ on the Polar Coordinate Plane

Graphing r = θ on the Polar Coordinate Plane To sketch the graph of r = θ, we should find some ordered pairs that satisfy the function. Since this function involves polar coordinates, we are looking for pairs of the form (r, θ).

Graphing r = θ on the Polar Coordinate Plane To sketch the graph of r = θ, we should find some ordered pairs that satisfy the function. Since this function involves polar coordinates, we are looking for pairs of the form (r, θ). Let’s construct a table so that we can find ordered pairs on the graph of r = θ.

Graphing r = θ on the Polar Coordinate Plane To construct a table of values for r = θ, let’s choose values for θ and find the corresponding values of r, and then form ordered pairs (r, θ).

Graphing r = θ on the Polar Coordinate Plane To construct a table of values for r = θ, let’s choose values for θ and find the corresponding values of r, and then form ordered pairs (r, θ). We can then plot these ordered pairs on a polar coordinate plane to obtain a graph of the function.

Graphing r = θ on the Polar Coordinate Plane Let’s start by choosing θ = 0.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) Let’s start by choosing θ = 0.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) Let’s start by choosing θ = 0. If θ = 0, then since r = θ we see that r also equals 0.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) Let’s start by choosing θ = 0. If θ = 0, then since r = θ we see that r also equals 0.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) Let’s start by choosing θ = 0. If θ = 0, then since r = θ we see that r also equals 0. This gives us the ordered pair (0, 0).

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) Let’s start by choosing θ = 0. If θ = 0, then since r = θ we see that r also equals 0. This gives us the ordered pair (0, 0).

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) Let’s start by choosing θ = 0. If θ = 0, then since r = θ we see that r also equals 0. This gives us the ordered pair (0, 0). Let’s plot this ordered pair on our polar coordinate plane.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) Now let’s choose another value for θ.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 Now let’s choose another value for θ. If θ = , π 4

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 0.8. π 4 π 4

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 4 ≈ 0.8 Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 0.8. π 4 π 4

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 4 ≈ 0.8 Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 0.8. This gives us the ordered pair . π 4 π 4 π 4 ( ) 0.8,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 4 ≈ 0.8 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 0.8. This gives us the ordered pair . π 4 π 4 π 4 ( ) 0.8,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 4 ≈ 0.8 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 0.8. This gives us the ordered pair . Let’s plot this ordered pair on our polar coordinate plane, π 4 π 4 π 4 ( ) 0.8,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 4 ≈ 0.8 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 0.8. This gives us the ordered pair . Let’s plot this ordered pair on our polar coordinate plane, and connect the point we already graphed with this new point. π 4 π 4 π 4 ( ) 0.8,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 4 ≈ 0.8 π 4 ( ) 0.8, Now let’s choose another value for θ.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , π 2

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 1.6. π 2 π 2

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 1.6. π 2 π 2

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 1.6. This gives us the ordered pair . π 2 π 2 π 2 ( ) 1.6,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 1.6. This gives us the ordered pair . π 2 π 2 π 2 ( ) 1.6,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 1.6. This gives us the ordered pair . Let’s plot this ordered pair on our polar coordinate plane, π 2 π 2 π 2 ( ) 1.6,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 1.6. This gives us the ordered pair . Let’s plot this ordered pair on our polar coordinate plane, and connect it with what we’ve graphed so far. π 2 π 2 π 2 ( ) 1.6,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , 3π 4

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 2.4. 3π 4 3π 4

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 2.4. 3π 4 3π 4

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 2.4. This gives us the ordered pair . 3π 4 3π 4 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 2.4. This gives us the ordered pair . 3π 4 3π 4 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 2.4. This gives us the ordered pair . Let’s plot this ordered pair on our polar coordinate plane, 3π 4 3π 4 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, Now let’s choose another value for θ. If θ = , then since r = θ we see that r also equals , which is approximately 2.4. This gives us the ordered pair . Let’s plot this ordered pair on our polar coordinate plane, and connect it with what we’ve graphed so far. 3π 4 3π 4 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, We can continue to choose values for θ,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, We can continue to choose values for θ, find the corresponding values for r,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, We can continue to choose values for θ, find the corresponding values for r, plot the ordered pair (r, θ) on our polar coordinate plane,

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, We can continue to choose values for θ, find the corresponding values for r, plot the ordered pair (r, θ) on our polar coordinate plane, and connect our points to obtain a graph of r = θ.

Graphing r = θ on the Polar Coordinate Plane r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, (

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, (

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, (

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, (

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( ( ) 3π 2 4.7,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( ( ) 3π 2 4.7,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( 3π 2 ( ) 4.7,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 7π 4 ≈ 5.5 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( 3π 2 ( ) 4.7,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 7π 4 ≈ 5.5 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( 3π 2 ( ) 4.7, ) 7π 4 5.5, (

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 7π 4 ≈ 5.5 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( ( ) ) 7π 4 5.5, ( 3π 2 4.7,

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 7π 4 ≈ 5.5 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( ( ) ) 7π 4 5.5, ( 3π 2 4.7, Finally, we can continue drawing our graph along this same trajectory.

Graphing r = θ on the Polar Coordinate Plane π r π ≈ 3.1 (r, θ ) (0, 0 ) (3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4 ≈ 0.8 π 2 ≈ 1.6 3π 4 ≈ 2.4 5π 4 ≈ 3.9 3π 2 ≈ 4.7 7π 4 ≈ 5.5 π 4 ( ) 0.8, π 2 ( ) 1.6, 3π 4 ( ) 2.4, ) 5π 4 3.9, ( ( ) ) 7π 4 5.5, ( 3π 2 4.7, Finally, we can continue drawing our graph along this same trajectory. The graph of r = θ is called the Archimedean Spiral.