Some Unit 5 Stuff

Parametric Functions

Represent curves where x and y depend on a parameter t; useful for motion problems; remember x=f(t) and y=g(t)

Parametric Modeling

Use parametric equations to model real-world scenarios like projectile motion; identify variables and constraints

Parametric Functions

Represent curves where x and y depend on a parameter t; useful for motion problems; remember x=f(t) and y=g(t)

Parametric Modeling

Use parametric equations to model real-world scenarios like projectile motion; identify variables and constraints

Parametric Circles

A circle can be parameterized as x=h+rcos(t), y=k+rsin(t); t is the angle in radians; remember the center (h,k) and radius r.

Conics

Recognize standard forms of parabolas, ellipses, and hyperbolas; parametrize conics (e.g., ellipse: x=acos(t), y=bsin(t)); know geometric properties like foci and directrix

Parameterization of Implicit Functions

Rewrite implicit equations in parametric form; use trigonometric identities or substitutions to simplify; verify the parameterization satisfies the original equation

Vectors

Represent quantities with magnitude and direction; add/subtract vectors component-wise; scalar multiplication scales magnitude; dot product measures angle between vectors

Matrices

Rectangular arrays of numbers; perform addition, subtraction, and scalar multiplication element-wise; matrix multiplication follows row-by-column rule

Inverse and Determinant of a Matrix

For a square matrix A, A^−1 exists if det(A) =/= 0

Linear Transformations and Matrices

Matrices represent transformations like rotations, reflections, and scalings; apply transformation by multiplying matrix by vector

Matrices as Functions

Treat matrices as functions mapping input vectors to output vectors; interpret columns of matrix as images of basis vectors under transformation.