$\oint 2.1$ Relations and Functions

Definition: A relation is any set of orderer pairs

Definition: A function is a specific relation where there is exactly one dependent variable for each independent (Vertical Line Test)

$\oint 2.2$ Slope and Rate of Change

Parallel: $l_1 \parallel l_2 \to m_1 = m_2$

Perpendicular $l_1 \bot l_2 \to m_1 = \frac{-1}{m_2}$

$\oint 2.3$ Graphing Lines

$\oint 2.4$ Equations of Lines

Slope Intercept Form: $y = mx + b$

Standard Form: $Ax + By = C, \text{where } A,B \neq 0, \text{and } A,B,C \in \mathbb Z$

Point-Slope Form: $y-y_1 = m(x-x_1)$

Common Problems:

  • Given a domain and a range, sketch a graph

Piecewise Functions

$f(x)= \begin{cases} \text{use this equation} & \text{when this condition is met} \\ \end{cases} $

$h(x)= \begin{cases} x + 1 & x\leq 0 \\ x^2 + 1 & x \gt 0 \\ \end{cases} $

$\oint 2.5$ Direct Variation

y varies directly as x

$y=kx$

y varis indirectly as x

$y=\frac{k}{x}$

$\oint 2.6$ Scatter Plots and Linear Regression

Correlation $\neq$ Causation

i.e. Ice Cream sales correlate with city violence, but they do not cause it because of a confounding variable, which is the heat

Correlation Coefficient: $r$

The closer $r$ is to $\pm 1$, the better the line fits the data

Linear Regression (Least Squares Method) using a calculator

  1. Turn on STAT PLOT
  2. Enter data into $L_1$ and $L_2$
  3. Find equation under STAT -> CALC
  4. Adjust window (or use zoom 9)

$\oint 2.7$ Absolute Value Functions and Transformations

$f(x) = | x |$

$g(x) = 2 |x| \to$ Vertical Stretch

$g(x) = 4 - 2 |x| \to$ Up 4 and reflected down

$g(x) = |x+3| \to$ Shift left 3

$g(x) = |3x-16| \to$ Horizontal shift and compression

Transformations

$\displaystyle g(x) = a[f(bx-h)]+k$

where $a$ = vertical stretch/compression

$b$ = horizontal stretch/compression

$h$ = horizontal shift

$k$ = vertical shift

$\oint 2.8$ Linear Inequalities with Two Variables

Ex: $y<2x+3$

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