Step by step
The question
Question: graph r = 2 + 4 cos θ.
θ is the angle we turn, r is how far we go out.
1 Check the symmetry
cos of minus θ is the same as cos θ.
So turning up or down by the same angle gives the same r: the graph is its own mirror image in the x-axis.
So θ from 0° to 180° is enough: the rest is the mirror.
2 Make a table
A table: 7 angles, 30° apart, from 0° to 180°.
The angles go along the top.
cos of each angle, from the unit circle, to 2 decimal places: 1, 0.87, 0.5, and so on.
Then r = 2 plus 4 times that.
For 0°: 2 + 4 × 1 is 6.
3 Plot the points
The polar grid: circles 1, 2, 3 around the center, and a line for every 30°.
θ = 0°: turn 0°, then go out 6.
θ = 30°: turn 30°, go out 5.5.
And so on.
At 150°, r is negative: −1.5.
We turn 150°, then go 1.5 the other way, backwards through the center.
Careful!
A negative r is not nothing.
We go backwards, through the center, to the other side.
4 Join and mirror
Join the points in order, with a smooth curve.
The other half is the mirror image in the x-axis.
5 Name it
2 is smaller than 4: r turns negative for some angles, and the curve makes a small loop inside.
So it is a limaçon with an inner loop.
6 Check
Let's check one angle we did not use: θ = 270°.
cos 270° is 0.
So r is 2: turn 270°, go out 2.
The point is on our curve.
Right!
Let's look back.
First: check the symmetry.
Then: make a table.
Then: plot the points.
Then: join and mirror.
Then: name it.
Last: check.