All Flashcards
Explain how to parametrize y = f(x).
Set x(t) = t, then y(t) = f(t). 't' moves along the x-axis, and f(t) gives the corresponding y-value.
Explain how to parametrize x = f⁻¹(y).
Set y(t) = t, then x(t) = f⁻¹(t). 't' moves along the y-axis, and f⁻¹(t) gives the corresponding x-value.
How do 'a' and 'b' affect the shape of an ellipse in its parametric equations?
'a' stretches the ellipse along the x-axis, and 'b' stretches it along the y-axis.
Explain the difference between parametrizing a horizontal and a vertical hyperbola.
Horizontal: x(t) uses sec(t), y(t) uses tan(t). Vertical: x(t) uses tan(t), y(t) uses sec(t).
Why is it important to check the domain of 't' when parametrizing a curve?
The domain of 't' determines which portion of the curve is represented by the parametrization. Incorrect domain can lead to an incomplete or incorrect representation.
What is the significance of h and k in the parametric equations of conic sections?
h and k represent the x and y coordinates of the center of the conic section, respectively.
How does parametrization simplify the study of curves?
It allows us to describe complex curves using simpler functions of a single variable, making analysis and computation easier.
Explain how to parametrize a parabola.
Solve the parabola's equation for either x or y, then let the other variable equal t.
What is the domain for 't' when parametrizing an ellipse to trace the entire ellipse once?
What is the key idea to check if a parametrization is valid?
When you plug x(t) and y(t) into the original equation of the curve, it should always be true for all values of 't' in the domain.
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: . Semi-circle: .
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)?