Parametrization of a circle with radius 1 centered at (0,0)?

$x(t) = \cos(t), y(t) = \sin(t)$

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Parametrization of a circle with radius 1 centered at (0,0)?

$x(t) = \cos(t), y(t) = \sin(t)$

Parametrization of a function y = f(x)?

$x(t) = t, y(t) = f(t)$

Parametrization of an inverse function x = f⁻¹(y)?

$x(t) = f^{-1}(t), y(t) = t$

Parametric equations for an ellipse centered at (h, k)?

$x(t) = h + a\cos(t), y(t) = k + b\sin(t)$

Parametric equations for a horizontal hyperbola centered at (h, k)?

$x(t) = h + a\sec(t), y(t) = k + b\tan(t)$

Parametric equations for a vertical hyperbola centered at (h, k)?

$x(t) = h + a\tan(t), y(t) = k + b\sec(t)$

What is the equation of an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b?

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

What is the equation of a horizontal hyperbola centered at (h,k)?

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

What is the equation of a vertical hyperbola centered at (h,k)?

$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

What is the relationship between tan(t), sin(t), and cos(t)?

$\tan(t) = \frac{\sin(t)}{\cos(t)}$

Horizontal vs. Vertical Hyperbola Parametrization?

Horizontal: x(t) uses sec(t). Vertical: y(t) uses sec(t).

Parametrizing y = f(x) vs. x = f⁻¹(y)?

y = f(x): x(t) = t. x = f⁻¹(y): y(t) = t.

Ellipse vs. Circle Parametrization?

Circle: x(t) = r cos(t), y(t) = r sin(t). Ellipse: x(t) = h + a cos(t), y(t) = k + b sin(t).

Parametric vs. Cartesian equations?

Parametric: x and y are functions of t. Cartesian: y is a function of x (or vice-versa).

Parametrization of a parabola opening up/down vs. left/right?

Up/Down: y = f(x), x(t) = t. Left/Right: x = f(y), y(t) = t.

What is the difference between the equations for an ellipse and a hyperbola?

Ellipse: Addition between the squared terms. Hyperbola: Subtraction between the squared terms.

What are the differences between sine and secant functions?

Sine: Bounded between -1 and 1. Secant: Greater than or equal to 1, or less than or equal to -1.

What are the differences between tangent and cotangent functions?

Tangent: sin(t)/cos(t). Cotangent: cos(t)/sin(t).

How do the parameters 'a' and 'b' influence the shape of an ellipse versus a hyperbola?

Ellipse: 'a' and 'b' define the semi-major and semi-minor axes. Hyperbola: 'a' and 'b' relate to the distance from the center to the vertices and the shape of the asymptotes.

How does the domain of 't' differ when parametrizing a full circle versus a semi-circle?

Full circle: $0 \le t < 2\pi$ . Semi-circle: $0 \le t < \pi$ .

What is parametrization?

Expressing a curve's coordinates using a single variable, 't', as x(t) and y(t).

What does the parameter 't' represent in parametrization?

A variable that determines the position on a curve, like a GPS coordinate.

What is an implicit function?

A function where the dependent variable is not explicitly isolated on one side of the equation.

What is a conic section?

A curve obtained as the intersection of a plane with a cone. Examples: circle, ellipse, parabola, hyperbola.

What is the significance of the domain of 't' in parametrization?

It defines the portion of the curve that is traced by the parametric equations.

Define semi-major axis.

The longest radius of an ellipse, from the center to the farthest point on the ellipse.

Define semi-minor axis.

The shortest radius of an ellipse, from the center to the closest point on the ellipse.

What is a horizontal hyperbola?

A hyperbola that opens to the left and right.

What is a vertical hyperbola?

A hyperbola that opens up and down.

What is the relationship between sec(t) and cos(t)?

$\sec(t) = \frac{1}{\cos(t)}$

All Flashcards

Parametrization of a circle with radius 1 centered at (0,0)?

$x(t) = \cos(t), y(t) = \sin(t)$

Parametrization of a function y = f(x)?

$x(t) = t, y(t) = f(t)$

Parametrization of an inverse function x = f⁻¹(y)?

$x(t) = f^{-1}(t), y(t) = t$

Parametric equations for an ellipse centered at (h, k)?

$x(t) = h + a\cos(t), y(t) = k + b\sin(t)$

Parametric equations for a horizontal hyperbola centered at (h, k)?

$x(t) = h + a\sec(t), y(t) = k + b\tan(t)$

Parametric equations for a vertical hyperbola centered at (h, k)?

$x(t) = h + a\tan(t), y(t) = k + b\sec(t)$

What is the equation of an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b?

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

What is the equation of a horizontal hyperbola centered at (h,k)?

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

What is the equation of a vertical hyperbola centered at (h,k)?

$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

What is the relationship between tan(t), sin(t), and cos(t)?

$\tan(t) = \frac{\sin(t)}{\cos(t)}$

Horizontal vs. Vertical Hyperbola Parametrization?

Horizontal: x(t) uses sec(t). Vertical: y(t) uses sec(t).

Parametrizing y = f(x) vs. x = f⁻¹(y)?

y = f(x): x(t) = t. x = f⁻¹(y): y(t) = t.

Ellipse vs. Circle Parametrization?

Circle: x(t) = r cos(t), y(t) = r sin(t). Ellipse: x(t) = h + a cos(t), y(t) = k + b sin(t).

Parametric vs. Cartesian equations?

Parametric: x and y are functions of t. Cartesian: y is a function of x (or vice-versa).

Parametrization of a parabola opening up/down vs. left/right?

Up/Down: y = f(x), x(t) = t. Left/Right: x = f(y), y(t) = t.

What is the difference between the equations for an ellipse and a hyperbola?

Ellipse: Addition between the squared terms. Hyperbola: Subtraction between the squared terms.

What are the differences between sine and secant functions?

Sine: Bounded between -1 and 1. Secant: Greater than or equal to 1, or less than or equal to -1.

What are the differences between tangent and cotangent functions?

Tangent: sin(t)/cos(t). Cotangent: cos(t)/sin(t).

How do the parameters 'a' and 'b' influence the shape of an ellipse versus a hyperbola?

Ellipse: 'a' and 'b' define the semi-major and semi-minor axes. Hyperbola: 'a' and 'b' relate to the distance from the center to the vertices and the shape of the asymptotes.

How does the domain of 't' differ when parametrizing a full circle versus a semi-circle?

Full circle: $0 \le t < 2\pi$ . Semi-circle: $0 \le t < \pi$ .

What is parametrization?

Expressing a curve's coordinates using a single variable, 't', as x(t) and y(t).

What does the parameter 't' represent in parametrization?

A variable that determines the position on a curve, like a GPS coordinate.

What is an implicit function?

A function where the dependent variable is not explicitly isolated on one side of the equation.

What is a conic section?

A curve obtained as the intersection of a plane with a cone. Examples: circle, ellipse, parabola, hyperbola.

What is the significance of the domain of 't' in parametrization?

It defines the portion of the curve that is traced by the parametric equations.

Define semi-major axis.

The longest radius of an ellipse, from the center to the farthest point on the ellipse.

Define semi-minor axis.

The shortest radius of an ellipse, from the center to the closest point on the ellipse.

What is a horizontal hyperbola?

A hyperbola that opens to the left and right.

What is a vertical hyperbola?

A hyperbola that opens up and down.

What is the relationship between sec(t) and cos(t)?

$\sec(t) = \frac{1}{\cos(t)}$