Graphing r on the Polar Coordinate Plane Graphing

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 θ r (r, θ ) 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 θ r (r, θ ) 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 θ r (r, θ ) Let’s start by choosing θ = 0.

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

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

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

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) Now let’s choose another value for θ. If θ = π , 4 then since r = θ we see that r also equals π , 4 which is approximately 0. 8.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 ≈ 0. 8 (r, θ ) (0, 0 ) Now let’s choose another value for θ. If θ = π , 4 then since r = θ we see that r also equals π , 4 which is approximately 0. 8.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 ≈ 0. 8 (r, θ ) (0, 0 ) Now let’s choose another value for θ. If θ = π , 4 then since r = θ we see that r also equals π , 4 which is approximately 0. 8. ( This gives us the ordered pair 0. 8, π 4 ).

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( ≈ 0. 8, π 4 ) Now let’s choose another value for θ. If θ = π , 4 then since r = θ we see that r also equals π , 4 which is approximately 0. 8. ( This gives us the ordered pair 0. 8, π 4 ).

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( ≈ 0. 8, π 4 ) Now let’s choose another value for θ. If θ = π , 4 then since r = θ we see that r also equals π , 4 which is approximately 0. 8. ( This gives us the ordered pair 0. 8, π 4 ). Let’s plot this ordered pair on our polar coordinate plane,

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( ≈ 0. 8, π 4 ) Now let’s choose another value for θ. If θ = π , 4 then since r = θ we see that r also equals π , 4 which is approximately 0. 8. ( This gives us the ordered pair 0. 8, π 4 ). Let’s plot this ordered pair on our polar coordinate plane, and connect the point we already graphed with this new point.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( ≈ 0. 8, π 4 ) Now let’s choose another value for θ.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 ) Now let’s choose another value for θ. If θ = π , 2

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 ) Now let’s choose another value for θ. If θ = π , 2 then since r = θ we see that r also equals π , 2 which is approximately 1. 6.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 ≈ 1. 6 ) Now let’s choose another value for θ. If θ = π , 2 then since r = θ we see that r also equals π , 2 which is approximately 1. 6.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 ≈ 1. 6 ) Now let’s choose another value for θ. If θ = π , 2 then since r = θ we see that r also equals π , 2 which is approximately 1. 6. ( This gives us the ordered pair 1. 6, π 2 ).

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 ≈ 1. 6 )( 1. 6, π 2 ) Now let’s choose another value for θ. If θ = π , 2 then since r = θ we see that r also equals π , 2 which is approximately 1. 6. ( This gives us the ordered pair 1. 6, π 2 ).

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 ≈ 1. 6 )( 1. 6, π 2 ) Now let’s choose another value for θ. If θ = π , 2 then since r = θ we see that r also equals π , 2 which is approximately 1. 6. ( This gives us the ordered pair 1. 6, π 2 ). Let’s plot this ordered pair on our polar coordinate plane,

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 ≈ 1. 6 )( 1. 6, π 2 ) Now let’s choose another value for θ. If θ = π , 2 then since r = θ we see that r also equals π , 2 which is approximately 1. 6. ( This gives us the ordered pair 1. 6, π 2 ). Let’s plot this ordered pair on our polar coordinate plane, and connect it with what we’ve graphed so far.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( ≈ 1. 6, π 2 ) Now let’s choose another value for θ.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 ) Now let’s choose another value for θ. If θ = 3π , 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 ) Now let’s choose another value for θ. If θ = 3π , 4 then since r = θ we see that r also equals 3π , 4 which is approximately 2. 4.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 ≈ 2. 4 ) Now let’s choose another value for θ. If θ = 3π , 4 then since r = θ we see that r also equals 3π , 4 which is approximately 2. 4.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 3π 4 ≈ 1. 6 )( 1. 6, π 2 3π 4 ≈ 2. 4 ) Now let’s choose another value for θ. If θ = 3π , 4 then since r = θ we see that r also equals 3π , 4 which is approximately 2. 4. ( This gives us the ordered pair 2. 4, ). 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 3π 4 ≈ 1. 6 )( 1. 6, π 2 3π 4 ≈ 2. 4 )( 2. 4, Now let’s choose another value for θ. If θ = 3π , 4 then since r = θ we see that r also equals 3π , 4 which is approximately 2. 4. ( This gives us the ordered pair 2. 4, ). 3π 4 ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 3π 4 ≈ 1. 6 )( 1. 6, π 2 3π 4 ≈ 2. 4 )( 2. 4, Now let’s choose another value for θ. If θ = 3π , 4 then since r = θ we see that r also equals 3π , 4 which is approximately 2. 4. ( This gives us the ordered pair 2. 4, ). 3π 4 Let’s plot this ordered pair on our polar coordinate plane, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 π 4 (r, θ ) (0, 0 ) ( π 2 ≈ 0. 8, π 4 π 2 3π 4 ≈ 1. 6 )( 1. 6, π 2 3π 4 ≈ 2. 4 )( 2. 4, Now let’s choose another value for θ. If θ = 3π , 4 then since r = θ we see that r also equals 3π , 4 which is approximately 2. 4. ( This gives us the ordered pair 2. 4, ) 3π 4 ). 3π 4 Let’s plot this ordered pair on our polar coordinate plane, and connect it with what we’ve graphed so far.

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 ≈ 2. 4 )( 2. 4, We can continue to choose values for θ, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 ≈ 2. 4 )( 2. 4, We can continue to choose values for θ, find the corresponding values for r, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 ≈ 2. 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, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 ≈ 2. 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. ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( π ≈ 2. 4, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( π ≈ 2. 4 π ≈ 3. 1 2. 4, ) 3π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( π ≈ 2. 4 π ≈ 3. 1 2. 4, ) 3π 4 (3. 1, π)

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( π ≈ 2. 4 π ≈ 3. 1 2. 4, ) 3π 4 (3. 1, π)

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( π ≈ 2. 4 π ≈ 3. 1 2. 4, ) 3π 4 (3. 1, π) 5π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ≈ 3. 9

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( ≈ 3. 9, ) 5π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( ≈ 3. 9, ) 5π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, ) 5π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, ) 5π 4 3π 2 ≈ 4. 7

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, 3π 2 )( 5π 4 ≈ 4. 7, ) 3π 2

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, 3π 2 )( 5π 4 ≈ 4. 7, ) 3π 2

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, 3π 2 )( 5π 4 7π 4 ≈ 4. 7, ) 3π 2

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, 3π 2 )( 5π 4 7π 4 ≈ 4. 7, ) 3π 2 7π 4 ≈ 5. 5

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 )( 3π 4 ≈ 1. 6, π 2 3π 4 )( Finally, we can continue drawing our graph along this same trajectory. ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, 3π 2 )( 5π 4 7π 4 ≈ 4. 7, 7π 4 )( 3π 2 ≈ 5. 5, ) 7π 4

Graphing r = θ on the Polar Coordinate Plane θ 0 r 0 (r, θ ) (0, 0 ) π 4 ( π 2 ≈ 0. 8, π 4 π 2 3π 4 ≈ 1. 6 )( 1. 6, π 2 )( Finally, we can continue drawing our graph along this same trajectory. The graph of r = θ is called the Archimedean Spiral. 3π 4 ≈ 2. 4 π ≈ 3. 1 2. 4, 5π 4 π ) 3π 4 (3. 1, π) 5π 4 ( 3π 2 ≈ 3. 9, 3π 2 )( 5π 4 7π 4 ≈ 4. 7, 7π 4 )( 3π 2 ≈ 5. 5, ) 7π 4