All Flashcards
What are the differences between Cartesian and Polar coordinates?
Cartesian: Uses (x, y), rectangular grid | Polar: Uses (r, θ), circular grid.
Compare the graphs of y = x and r = θ.
y = x: Straight line | r = θ: Spiral.
Compare the equations for a circle in Cartesian and Polar form.
Cartesian: (x-h)² + (y-k)² = r² | Polar: r = a (centered at origin).
Compare symmetry about the x-axis in Cartesian and Polar coordinates.
Cartesian: Replace y with -y | Polar: Replace θ with -θ.
Compare symmetry about the y-axis in Cartesian and Polar coordinates.
Cartesian: Replace x with -x | Polar: Replace θ with π - θ.
Compare the effect of changing 'a' in r = a cos(θ) and r = cos(aθ).
r = a cos(θ): Changes diameter of circle | r = cos(aθ): Changes the number of 'petals'.
Compare the concept of periodicity in Cartesian and Polar functions.
Cartesian: f(x) = f(x + p) | Polar: f(θ) = f(θ + 2π) (typically).
Compare the representation of the origin in Cartesian and Polar coordinates.
Cartesian: Unique representation (0,0) | Polar: Infinite representations (0, θ) for any θ.
Compare graphing a line in Cartesian (y = mx + b) and Polar coordinates.
Cartesian: Straight line | Polar: Requires converting to polar equation, can be more complex.
Compare the area calculation for a region in Cartesian and Polar coordinates.
Cartesian: ∫∫ dxdy | Polar: ∫∫ r dr dθ
What does the graph of r = θ represent?
A spiral that extends outward from the origin as θ increases.
What shape does r = sin(θ) create?
A circle that touches the origin with a diameter along the y-axis.
What does the graph of r = 2cos(θ) represent?
A circle with diameter 2 along the x-axis, passing through the origin.
How can you identify symmetry about the x-axis from a polar graph?
The graph is symmetric about the x-axis if the portion above the x-axis is a mirror image of the portion below it.
How can you identify symmetry about the y-axis from a polar graph?
The graph is symmetric about the y-axis if the portion to the right of the y-axis is a mirror image of the portion to the left of it.
What does a cardioid graph look like?
A heart-shaped curve with a cusp at the origin.
What does a lemniscate graph look like?
A figure-eight-shaped curve centered at the origin.
How does the coefficient 'a' in r = a sin(θ) affect the circle's graph?
It determines the diameter of the circle. A larger 'a' means a larger diameter.
What does the graph of r = a (where a is a constant) represent?
A circle centered at the origin with radius 'a'.
What happens to the graph of r = f(θ) when 'r' values are negative?
The points are reflected through the origin, changing the direction of the radius.
Define polar coordinates.
A system using (r, θ) to locate points, where r is the distance from the origin and θ is the angle from the positive x-axis.
What is a polar function?
An equation in the form r = f(θ), where θ is the input angle and r is the output distance from the origin.
Define symmetry about the origin in polar functions.
A polar function is symmetric about the origin if its graph is unchanged when rotated by 180 degrees.
What is periodicity in polar functions?
A polar function is periodic if its graph repeats after a fixed interval of θ.
Define 'r' in polar coordinates.
The distance from the origin (0,0) to the point in the polar plane.
Define 'θ' in polar coordinates.
The angle measured counterclockwise from the positive x-axis to the point in the polar plane.
What is a cardioid?
A heart-shaped polar curve, often represented by the equation r = a(1 ± cos θ) or r = a(1 ± sin θ).
What is a lemniscate?
A figure-eight-shaped polar curve, often represented by the equation r² = a²cos(2θ) or r² = a²sin(2θ).
What does a negative 'r' value signify in polar coordinates?
It indicates a point located in the opposite direction of the angle θ from the origin.
What is the polar equation of a circle centered at the origin?
r = a, where 'a' is the radius of the circle.