1.7 Graphs of Polar Equations

How do you determine the symmetry of a polar graph?

• A polar graph is symmetric about the polar axis if replacing θ with -θ yields the same equation, and symmetric about the line @@ heta = rac{ ext{π}}{2}@@ if replacing θ with @@ ext{π} - heta@@ yields the same equation.

• A polar graph is symmetric about the origin if replacing r with -r yields the same equation.

• A polar graph is symmetric about the line @@ heta = 0@@ if replacing θ with @@rac{ ext{π}}{2}@@ yields the same equation.

• A polar graph is symmetric about the line @@ heta = rac{ ext{π}}{4}@@ if replacing θ with @@rac{ ext{3π}}{4}@@ yields the same equation.

What does the polar equation @@r = -2@@ represent?

• A line passing through the origin

• A circle centered at the origin with a radius of 2

• A point at the origin

• A circle centered at (2,0) with a radius of 2

What is the polar equation for a circle centered at the origin?

• @@r = a@@, where a is the radius

• @@r = 2a@@, where a is the radius

• @@r = a^2@@, where a is the radius

• @@r = rac{1}{a}@@, where a is the radius

What is the equation of a rose curve?

• @@r = a \times \text{sin}(n\theta)@@

• @@r = a \times \text{cos}(n\theta)@@

• @@r = a \times \text{tan}(n\theta)@@

• @@r = a \times n\theta@@

What does the polar equation @@r = 1 + \text{cos}(\theta)@@ represent?

• A circle

• A cardioid

• An ellipse

• A hyperbola

Define a cardioid in polar coordinates.

• A cardioid is a heart-shaped curve represented by the polar equation @@r = 1 + heta@@ or @@r = 1 - heta@@.

• A cardioid is a circular curve represented by the polar equation @@r = 2 + heta@@.

• A cardioid is a linear curve represented by the polar equation @@r = heta@@.

• A cardioid is a spiral curve represented by the polar equation @@r = heta^2@@.

What is the significance of the pole in polar coordinates?

• The pole is the origin point in polar coordinates, where the distance r is zero.

• The pole represents the maximum distance in polar coordinates.

• The pole is the point where all angles converge in polar coordinates.

• The pole is used to define the Cartesian coordinates in polar systems.

How can you identify a dimpled limacon?

• A dimpled limacon has a polar equation of the form @@r = a + b\text{cos}(\theta)@@ or @@r = a + b\text{sin}(\theta)@@, where a < b < 2a.

• A dimpled limacon is characterized by having a polar equation of the form @@r = a - b\text{cos}(\theta)@@ or @@r = a - b\text{sin}(\theta)@@, where a > b.

• A dimpled limacon has a polar equation of the form @@r = a + b\text{cos}(\theta)@@ or @@r = a + b\text{sin}(\theta)@@, where a = b.

• A dimpled limacon is identified by its rectangular equation of the form @@x^2 + y^2 = r^2@@.

What is the polar equation for a line passing through the origin?

What is a spiral of Archimedes?

• A spiral represented by the polar equation @@r = a + b\theta@@, where a and b are constants, and it spirals outward as θ increases.

• A circular motion defined by the equation @@r = a\theta@@, where a is a constant.

• A logarithmic spiral defined by the equation @@r = ae^{b\theta}@@, where a and b are constants.

• A spiral that remains constant in radius as it rotates.