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Horizontal vs. Vertical Hyperbola Parametrization?

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

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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≤t<2π0 \le t < 2\pi. Semi-circle: 0≤t<π0 \le t < \pi.

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

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

Parametrization of a function y = f(x)?

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

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

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

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

x(t)=h+acos⁡(t),y(t)=k+bsin⁡(t)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+asec⁡(t),y(t)=k+btan⁡(t)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+atan⁡(t),y(t)=k+bsec⁡(t)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?

(x−h)2a2+(y−k)2b2=1\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)?

(x−h)2a2−(y−k)2b2=1\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)?

(y−k)2a2−(x−h)2b2=1\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)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}

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?

0≤t<2π0 \le t < 2\pi

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.