Graphing r = θ on the Polar Coordinate Plane. To sketch the graph of r = θ, we should find some...
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Transcript of Graphing r = θ on the Polar Coordinate Plane. To sketch the graph of r = θ, we should find some...
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 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,(