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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

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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

(r, θ ) (0, 0 )

Now let’s choose another value for θ.

If θ = , π4

π4

π4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π4

π4

π4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π4

π4

π4

≈ 0.8

π4

π4

≈ 0.8

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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 .

π4

π4

π4( )0.8,

π4

π4

≈ 0.8

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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 .

π4

π4

π4( )0.8,

π4( )0.8,

π4

π4

≈ 0.8

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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,

π4

π4

π4( )0.8,

π4( )0.8,

π4

π4

≈ 0.8

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π4

π4

π4( )0.8,

π4( )0.8,

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

Now let’s choose another value for θ.

π4

π4( )0.8,

π4

≈ 0.8

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

Now let’s choose another value for θ.

If θ = , π2

π4

π4( )0.8,

π4

≈ 0.8

π2

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π2

π2

π4

π4( )0.8,

π4

≈ 0.8

π2

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π2

π2

π4

π4( )0.8,

π4

≈ 0.8

π2

π2

≈ 1.6

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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 .

π2

π2

π4

π4( )0.8,

π2( )1.6,

π4

≈ 0.8 π2

≈ 1.6

π2

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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 .

π2

π2

π4

π4( )0.8,

π2( )1.6,

π4

≈ 0.8

π2( )1.6,

π2

≈ 1.6

π2

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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,

π2

π2

π4

π4( )0.8,

π2( )1.6,

π4

≈ 0.8

π2( )1.6,

π2

≈ 1.6

π2

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π2

π2

π4

π4( )0.8,

π2( )1.6,

π4

≈ 0.8

π2

π2( )1.6,

π2

≈ 1.6

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

Now let’s choose another value for θ.

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

Now let’s choose another value for θ.

If θ = , 3π4

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

3π4

3π4

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

3π4

3π4

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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 .

3π4

3π4

π4

π4( )0.8,

3π4( )2.4,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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 .

3π4

3π4

π4

π4( )0.8,

3π4( )2.4,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4

≈ 2.4

3π4( )2.4,

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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,

3π4

3π4

π4

π4( )0.8,

3π4( )2.4,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4

≈ 2.4

3π4( )2.4,

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

3π4

3π4

π4

π4( )0.8,

3π4( )2.4,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4

≈ 2.4

3π4( )2.4,

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

We can continue to choose values for θ,

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

We can continue to choose values for θ, find the corresponding values for r,

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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,

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

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.

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0

r 0

(r, θ ) (0, 0 )

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0

(r, θ ) (0, 0 )

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 )

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4 5π4

≈ 3.9

5π4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

) 5π43.9,(

5π4

5π4

≈ 3.9

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

) 5π43.9,(

5π4

5π4

≈ 3.9

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π24.7,

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,( ( )

3π2

3π2

≈ 4.7

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

≈ 4.7

( ) 3π24.7,

3π2

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π2( )4.7,

7π4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π2( )4.7,

7π4

7π4

≈ 5.5

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π2( )4.7,

7π4

) 7π45.5,(

7π4

≈ 5.5

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π2( )4.7,

7π4

≈ 5.5

) 7π45.5,(

7π4

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

Finally, we can continue drawing our

graph along this same trajectory.

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π2( )4.7,

7π4

7π4

≈ 5.5

) 7π45.5,(

Graphing r = θ on the Polar Coordinate Plane

θ 0 π

r 0 π ≈ 3.1

(r, θ ) (0, 0 ) (3.1, π)

Finally, we can continue drawing our

graph along this same trajectory.

The graph of r = θ is called the

Archimedean Spiral.

π4

π4( )0.8,

π4

≈ 0.8 π2

≈ 1.6

π2

π2( )1.6,

3π4

3π4( )2.4,

3π4

≈ 2.4

5π4

5π4

≈ 3.9

) 5π43.9,(

3π2

3π2

≈ 4.7

3π2( )4.7,

7π4

7π4

≈ 5.5

) 7π45.5,(