Limaçon Equations and Polar Graphs

This video tutorial covers how to graph polar equations, focusing on circles and limaçons. It explains the equations for graphing circles using cosine and sine, detailing how the sign and value of 'a' affect the graph's direction and size. The tutorial also introduces limaçons, describing their different shapes based on the ratio of 'a' to 'b', and provides examples of graphing them with various parameters.

What happens to the circle in the polar equation r = a cos(θ) when 'a' is negative?

• The circle is directed towards the left.

• The circle is directed towards the right.

• The circle is centered at the origin.

• The circle becomes a line.

In the equation r = 4 cos(θ), what is the radius of the circle?

• 4

• 2

• 8

• 1

How is the area of a circle calculated using its radius in polar coordinates?

• 2πr

• πr²

• r²

• πr

For the equation r = a sin(θ), where is the circle centered when 'a' is positive?

• Above the x-axis

• At the origin

• On the y-axis

• On the x-axis

What is the shape of the graph when a/b = 1 in a limaçon equation?

• Limaçon with inner loop

• Cardioid

• Dimpled limaçon

• Circle

In a limaçon equation r = a ± b cos(θ), what does a positive cosine indicate?

• The graph opens downwards.

• The graph opens upwards.

• The graph opens towards the right.

• The graph opens towards the left.

What is the first step in graphing a limaçon with an inner loop?

• Determine the value of a/b.

• Find the y-intercepts.

• Calculate the radius.

• Identify the x-intercepts.

In the equation r = 3 + 5 cos(θ), what is the value of a/b?

• 0.8

• 2.0

• 1.5

• 0.6

What type of limaçon is formed when a/b > 2?

• Cardioid

• Limaçon without a dimple or inner loop

• Dimpled limaçon

• Limaçon with inner loop

In the equation r = 3 - 7 cos(θ), what direction does the graph open?

• Right

• Left

• Downwards

• Upwards