Functions Involving Parameters, Vectors, and Matrices

Notes

Parametric functions

A parametric function gives $x$ and $y$ separately in terms of a parameter $t$: $(x(t), y(t))$. Think of $t$ as time and the point as a moving particle.

• To graph, make a table of $t$, $x$ and $y$. Arrows show the direction of motion as $t$ increases.
• Eliminate the parameter by solving one equation for $t$ and substituting. Example: $x = t + 1$, $y = t^2$ becomes $y = (x - 1)^2$.
• A circle: $x = r\cos t$, $y = r\sin t$.
• Average rate of change of $x(t)$ on $[a, b]$ is $\displaystyle \frac{x(b) - x(a)}{b - a}$ (horizontal speed on average).

Conic sections

• Parabola: $y = a(x - h)^2 + k$.
• Ellipse: $\displaystyle \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$.
• Hyperbola: $\displaystyle \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$.

Vectors

A vector has magnitude and direction. $\vec v = \langle a, b \rangle$.

• Magnitude $\displaystyle |\vec v| = \sqrt{a^2 + b^2}$. Unit vector: $\displaystyle \frac{\vec v}{|\vec v|}$.
• Add componentwise. Scalar multiples stretch the vector.
• Dot product: $\langle a, b\rangle \cdot \langle c, d\rangle = ac + bd = |\vec u||\vec v|\cos\theta$. A dot product of 0 means the vectors are perpendicular.

Matrices

• Multiply an $m \times n$ matrix by an $n \times p$ matrix to get $m \times p$. Order matters.
• For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\det A = ad - bc$ and $\displaystyle A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ when $\det A \ne 0$.
• A matrix can represent a linear transformation of the plane. The absolute value of the determinant is the area scale factor.
• Transition matrices model states changing over time: multiply the current state vector by the matrix to get the next state.