Comprehensive Study Notes: Graphing Linear Equations, Midpoint, and Distance

Course Units: The course is essentially broken into three major units. The initial unit (Module 4) focuses on linear equations and graphing, though the instructor believes it would be better placed in later modules that specifically cover graphing.
Pre-Module 4 Topics: Previously, the course covered topics like the quadratic formula, square rooting, factoring, rational expressions, and radical expressions, largely without graphical representations.
Module 4 Significance: This module (Module 4) is the last before the first exam (Test 1).
Knowledge Checks: There are three knowledge checks for the term. The second knowledge check (Knowledge Check 2) is due the Saturday after the current week, tied to every two modules.
Test 1 Details:
- Date: Scheduled for October 7th.
- Preparation: A screening for Test 1 will be provided next week, possibly with a practice test in ALEKS.
- Question Count: The actual exam will have $20$ questions. The practice test in ALEKS is about double the length (around $40$ questions).
- Requirements: Requires a computer/server (not just ALEKS) and a lockdown browser (pre-installed).
- Allowed Aids: Students are permitted one full sheet of paper for formulas, examples, or notes.
Homework: Module 4 Course Green Homework is due by Tuesday; Knowledge Check 2 is due on Saturday; Test 1 is due the following Tuesday (October 2nd).

General Approach: Graphing linear equations is considered the easiest type of graph to make, compared to quadratics, rationals, radicals, or higher-order polynomials (e.g., $x^3$ ).
Methods for Graphing Straight Lines:
1. Using Two $(x, y)$ Coordinates:
  - Any straight line can be graphed with just two distinct points.
  - Process: Choose two arbitrary $x$ values, substitute them into the equation, and solve for the corresponding $y$ values to obtain two $(x, y)$ coordinate pairs.
  - Example (for $y = 3x - 5$ ):
    - If $x = -4$ , then $y = 3(-4) - 5 = -12 - 5 = -17$ . Plot $(-4, -17)$ .
    - If $x = -2$ , then $y = 3(-2) - 5 = -6 - 5 = -11$ . Plot $(-2, -11)$ .
  - Practicality: Manual graphing can be difficult, especially with large $y$ values requiring careful scaling (e.g., counting by ones vertically).
  - ALEKS Tip: ALEKS provides tools for plotting points, and in some cases, allows direct input of coordinates, including fractions.
2. Using Slope-Intercept Form ( $y = mx + b$ ):
  - This form directly provides two key pieces of information.
  - $b$ (y-intercept): Represents the point where the line crosses the $y$ -axis. This occurs when $x = 0$ . For $y = 3x - 5$ , the $y$ -intercept is $(0, -5)$ . Plot this point first.
  - $m$ (slope): Represents the