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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

Recognize as a circle. x(t) = 3cos(t), y(t) = 3sin(t).

Let x(t) = t, then y(t) = t³.

Let y(t) = t, then x(t) = t² + 1.

Circle centered at (1, -2) with radius 2. x(t) = 1 + 2cos(t), y(t) = -2 + 2sin(t).

Ellipse centered at (2, -1). x(t) = 2 + 3cos(t), y(t) = -1 + 4sin(t).

Vertical hyperbola. x(t) = 3tan(t), y(t) = 5sec(t).

Horizontal hyperbola centered at (1, -2). x(t) = 1 + 2sec(t), y(t) = -2 + 3tan(t).

Let x(t) = t. Then, y(t) = t² - 4t + 7.

Solve for t in x(t): t = (x-1)/2. Substitute into y(t): y = ((x-1)/2)² = (x-1)²/4.

x(t) = x₁ + t(x₂ - x₁), y(t) = y₁ + t(y₂ - y₁), 0 ≤ t ≤ 1