Quadratics, Linear Graphs, Parallel/Perpendicular Lines & Functions

Indicators A.4 through A.7 form the culmination of the GED Mathematical Reasoning test: solving quadratic equations in one variable (by factoring, completing the square, or the quadratic formula), interpreting key features of graphs (x- and y-intercepts, slope, increasing/decreasing intervals, maxima and minima), writing equations of lines (including parallel and perpendicular lines), and evaluating or comparing functions presented as equations, tables, graphs, or verbal descriptions. Once again, the official GED Formula Sheet is your ally: it prints the Slope formula m = (y₂ − y₁)/(x₂ − x₁), Slope-Intercept form y = mx + b, Point-Slope form y − y₁ = m(x − x₁), Standard form of a quadratic equation y = ax² + bx + c, and the full Quadratic Formula x = [−b ± √(b² − 4ac)] / (2a).

1. Solving Quadratic Equations ax² + bx + c = 0 (Indicator A.4): • Always move every term to one side so the equation equals 0 BEFORE factoring or reading a, b, and c! • Method 1 — Factoring & Zero-Product Property: If (x − 5)(x + 2) = 0, then x − 5 = 0 (so x = 5) or x + 2 = 0 (so x = −2). • Method 2 — Completing the Square: For x² + bx = d, add (b/2)² to both sides so the left side becomes the perfect square (x + b/2)² = d + (b/2)², then take ± square roots. • Method 3 — The Quadratic Formula: x = [−b ± √(b² − 4ac)] / (2a). Works for every quadratic equation! Remember to put parentheses around negative b when squaring b². 2. Graphing Lines & Interpreting Graph Features (Indicator A.5): • Intercepts: The x-intercept is where the graph crosses the horizontal x-axis (set y = 0); the y-intercept is where it crosses the vertical y-axis (set x = 0). • Increasing vs Decreasing Intervals: Read the graph from left to right. Where the curve rises from left to right, the function is increasing; where it falls from left to right, it is decreasing. • Maxima and Minima (Vertex): The highest peak on a downward-opening parabola (a < 0) is its maximum; the lowest valley on an upward-opening parabola (a > 0) is its minimum. The vertex occurs at x = −b / (2a). 3. Writing Equations of Lines & Parallel/Perpendicular Slopes (Indicator A.6): • Given two points (x₁, y₁) and (x₂, y₂): first compute slope m = (y₂ − y₁)/(x₂ − x₁), then plug m and one point into y = mx + b (or point-slope form y − y₁ = m(x − x₁)) to find b. • Parallel lines have equal slopes (m₁ = m₂); perpendicular lines have negative-reciprocal slopes (m₂ = −1 / m₁). 4. Functions & Comparing Representations (Indicator A.7): • Vertical Line Test / Definition: A relation is a function if every input x has EXACLY ONE output y (no repeated x-values with different y-values in a table or ordered-pair list). • Function Notation: f(4) means 'plug x = 4 into the rule for f and compute the output y'—it does NOT mean f multiplied by 4! • Comparing Representations: When comparing Function A (given as an equation like f(x) = 3x + 2) with Function B (given as a table of values or a graph), compute the same feature (e.g., slope/rate of change, y-intercept, or maximum height) for both before comparing.