Unit 2 Notes: Lines, Linear Applications, and Function Transformations

Flashcards covering linear equation forms, parallel and perpendicular slopes, linear cost, revenue, and profit functions, parent functions, and graph transformations.

Slope-intercept form

The linear equation form $y = m x + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

Standard form of a linear equation

The linear equation written in the form $A x + B y = C$.

Slope formula

The formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ used to calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$.

Point-slope formula

The formula $y - y_1 = m (x - x_1)$ used to find the equation of a line given a slope $m$ and a point $(x_1, y_1)$.

Parallel lines

Lines that have the same slope.

Perpendicular lines

Lines with slopes that are negative reciprocals of each other.

Slopes of Lines A, B, and C Graph

A graph example where Line A has an undefined slope, Line B has slope $m = 0$, and Line C has slope $m = -\frac{1}{3}$.

Cost function $C(x)$

A linear model $C(x) = m x + b$, where $m$ is the variable cost per item and $b$ is the fixed cost.

Revenue function $R(x)$

A linear model $R(x) = p x$, where $p$ is the price per unit.

Profit function $P(x)$

The total profit equation defined as $P(x) = R(x) - C(x)$.

Break-even point

The point at which profit is $0$, where total cost equals total revenue ($C(x) = R(x)$).

Basic Parent Functions Graph

A set of parent curves including quadratic $f(x) = x^2$, cubic $f(x) = x^3$, absolute value $f(x) = |x|$, square root $f(x) = \sqrt{x}$, cube root $f(x) = \sqrt[3]{x}$, and rational $f(x) = \frac{1}{x}$.

Vertical stretch or shrink $a f(x)$

A transformation where multiplying $f(x)$ by a factor $a$ stretches or shrinks the graph vertically by $a$ units.

Reflection about the x-axis ($R_x$)

A transformation represented by $-f(x)$, which flips the graph of $f(x)$ across the $x$-axis.

Reflection about the y-axis ($R_y$)

A transformation represented by $f(-x)$, which flips the graph of $f(x)$ across the $y$-axis.

Vertical shift $f(x) + k$

A transformation that shifts the graph of $f(x)$ up or down by $|k|$ units.

Horizontal shift $f(x - h)$

A transformation that shifts the graph of $f(x)$ left or right by $|h|$ units.

Asymptotes of $f(x) = \frac{1}{x}$

Lines that the curve approaches, given by vertical asymptote $x = 0$ and horizontal asymptote $y = 0$.

Perpendicular Line Worksheet Examples

Worked solutions showing how to derive equations for lines perpendicular to a given line and calculating break-even points for linear applications.

Graphing Transformations Examples

Worked examples demonstrating combined horizontal and vertical shifts on parent curves, such as $f(x) = \frac{1}{x - 1} + 2$.