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 @@ heta + 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{π}}{4} - heta@@ yields the same equation.

What is the relationship between polar and Cartesian coordinates?

• The relationship is given by the equations @@x = r\text{cos}(\theta)@@ and @@y = r\text{sin}(\theta)@@.

• The relationship is given by the equations @@x = r\text{sin}(\theta)@@ and @@y = r\text{cos}(\theta)@@.

• The relationship is given by the equations @@x = \theta\text{cos}(r)@@ and @@y = \theta\text{sin}(r)@@.

• The relationship is given by the equations @@x = r\text{tan}(\theta)@@ and @@y = r\text{sec}(\theta)@@.

What is a polar curve?

• A graph that shows the relationship between distance and angle in polar coordinates.

• A type of graph that only represents linear equations.

• A graph that represents the distance from the origin as a function of time.

• A curve that can only be represented in Cartesian coordinates.

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 type of spiral that only exists in three-dimensional space.

• A spiral that is defined by a constant radius from a central point.

• A spiral that decreases in size as it rotates around a point.

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 @@1<\frac{a}{b}<2@@

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

• 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 characterized by having a constant radius in polar coordinates.

What is the equation of a rose curve?

• @@r=a\text{ sin}(n\theta)@@

• @@r=a\text{ cos}(n\theta)@@

• @@r=a\text{ tan}(n\theta)@@

• @@r=a\text{ sec}(n\theta)@@

What type of symmetry does the polar graph @@r = ext{sin}(2 heta)@@ exhibit?

• Symmetry about the x-axis

• Symmetry about the y-axis

• Symmetry about the origin

• Symmetry about the line y = x

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

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

• A circle

• A cardioid

• An ellipse

• A hyperbola

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.