Quadratic Functions (Copy)

QUADRATIC FUNCTIONS – Quick Revision Cheat Sheet

2.1 Finding Maximum or Minimum Value

• General form: f(x) = ax² + bx + c
  • If a > 0 → parabola opens upward → minimum point
  • If a < 0 → parabola opens downward → maximum point
• Methods:
  1. Completing the square:
     f(x) = a(x + p)² + q
     Vertex = (−p, q) → min/max value = q
  2. Differentiation:
     f'(x) = 2ax + b = 0 → x = −b / 2a
     Substitute into f(x) for min/max value.

2.2 Using Maximum/Minimum Value

• To Sketch Graph:
  1. Find vertex (turning point)
  2. Find y-intercept (x = 0 → y = c)
  3. Find x-intercepts (f(x) = 0)
• To Find Range:
  • For a > 0: Range is [minimum value, ∞)
  • For a < 0: Range is (−∞, maximum value]

2.3 Conditions for Roots & Intersection

Discriminant: Δ = b² − 4ac

• Δ > 0 → two distinct real roots (curve cuts x-axis twice)
• Δ = 0 → one real root (curve touches x-axis, tangent)
• Δ < 0 → no real roots (curve does not cross x-axis)

Line y = mx + c₁ intersecting curve y = ax² + bx + c₂:

• Set ax² + bx + c₂ = mx + c₁ → rearrange to quadratic in x
• Apply discriminant rules for intersection conditions.

2.4 Solving Quadratic Equations

• Factorisation: ax² + bx + c = 0 → (px + q)(rx + s) = 0
• Quadratic Formula:
  x = (−b ± √(b² − 4ac)) / 2a
• Completing the Square: Rearrange to (x + p)² = k

2.5 Quadratic Inequalities

• Method 1: Graphical
  • Sketch the parabola
  • Identify regions above/below x-axis depending on inequality sign.
• Method 2: Algebraic
  1. Solve f(x) = 0 for roots r₁ and r₂
  2. Test intervals between/around roots to see sign of f(x)
• Solution Form Examples:
  • r₁ < x < r₂
  • x < r₁ or x > r₂

Quick Reference Table

Common Mistakes

• Forgetting to divide by a when completing the square.
• Mixing up inequality signs after multiplying by negative numbers.
• Using incorrect range notation (must match domain restrictions).
• Ignoring need to check sign of a for min/max identification.

QUADRATIC FUNCTIONS – VISUAL CHART

1. Graph Shape & Turning Point

2. Finding Turning Point

Formula: x = −b / 2a
Substitute into f(x) to get turning point (h, k)

Example: f(x) = 2x² − 4x + 1
x = −(−4) / (2×2) = 1
f(1) = 2(1)² − 4(1) + 1 = −1 → min point (1, −1)

3. Discriminant & Roots

4. Inequalities – Sign Diagram Method

Steps:

1. Solve f(x) = 0 to find roots r₁, r₂
2. Draw number line with r₁, r₂
3. Test intervals to find sign of f(x)
4. Select intervals matching inequality

Example: f(x) = x² − 5x + 6 > 0
Roots: 2, 3
Sign pattern (a > 0 → + − +)
Answer: x < 2 or x > 3

5. Quick Visual Reference

For a > 0:

     +
    / 
   /   
  +     +

For a < 0:

  -     -
      /
     /
     -

(Signs above/below axis show f(x) sign in each region)

6. Common Mistakes Table