📋 Prerequisites

• Linear Equations & Inequalities

Functions are the bridge between algebra and geometry. A function takes an input and gives exactly one output — and when you graph all the (input, output) pairs, you get a picture. Understanding functions and their graphs unlocks coordinate geometry, calculus, and most of advanced math.

📚 Key Concepts

What Is a Function?

A function $f$ is a rule that assigns each input $x$ to exactly one output $f(x)$.

• Domain: All valid inputs (x-values)
• Range: All possible outputs (y-values)
• Vertical Line Test: A graph is a function if no vertical line crosses it more than once

Notation: $f(x)$ means "f of x" — it's the output when you plug $x$ into function $f$. It does NOT mean $f \times x$.

Linear Functions: $y = mx + b$

Slope-intercept form, where:

• $m$ (slope): $ \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} $ — steepness & direction
• $b$ (y-intercept): Where the line crosses the y-axis (when $x = 0$)

Positive slope goes up to the right; negative slope goes down to the right. Zero slope = horizontal line. Undefined slope = vertical line (not a function!)

Parallel & Perpendicular Slopes

• Parallel lines: Same slope, different intercepts. $ m_1 = m_2 $
• Perpendicular lines: Slopes are negative reciprocals. $ m_1 \times m_2 = -1 $
• Example: if one line has slope $ \frac{2}{3} $, a perpendicular line has slope $ -\frac{3}{2} $
• Special case: horizontal ($m=0$) and vertical ($m$ undefined) are perpendicular

Point-Slope Form

When you know a point $(x_1, y_1)$ and the slope $m$:

\[ y - y_1 = m(x - x_1) \]

This is often faster than slope-intercept for competition problems. You can always rearrange to $y = mx + b$ later.

Function Transformations (Shifts)

Given a parent function $f(x)$:

• $ f(x) + k $ — shift up by $k$ (vertical shift)
• $ f(x) - k $ — shift down by $k$
• $ f(x - h) $ — shift right by $h$ (horizontal shift, counterintuitive!)
• $ f(x + h) $ — shift left by $h$
• $ -f(x) $ — reflect over the x-axis (flip vertically)
• $ f(-x) $ — reflect over the y-axis (flip horizontally)
• $ a \cdot f(x) $ — vertical stretch ($a > 1$) or compression ($0 < a < 1$)

Absolute Value Function

$ f(x) = |x| $ makes a V-shape with vertex at (0, 0).

General form: $ y = a|x - h| + k $

• Vertex at $(h, k)$
• $a$ is the slope of each arm (positive opens up, negative opens down)
• Domain: all real numbers; Range: $ y \ge k $ (if $a > 0$)

✏️ Example Problems