Sketching Curves (Copy)

Graphs of Functions Cheat Sheet (IGCSE Mathematics 0580 – CORE)
Topic: Recognising, Sketching & Interpreting Graphs of Linear and Quadratic Functions

1. Linear Functions: y = ax + b

Shape:

• Straight line
• Extends in both directions infinitely
• No curve, no turning point

Key Features:

• Gradient = a
  • Positive → line slopes upwards
  • Negative → line slopes downwards
• y-intercept = b
  • The point where the line crosses the y-axis

General Rules:

• x-intercept: set y = 0 and solve for x
• y-intercept: set x = 0 → y = b

Example:

y = 2x – 3

• Gradient = 2 (line rises steeply)
• y-intercept = –3 (crosses y-axis at –3)

2. Quadratic Functions: y = ax² + bx + c

Shape:

• Parabola (U-shaped or ∩-shaped curve)
• Opens upwards if a > 0 (U-shape)
• Opens downwards if a < 0 (∩-shape)

Key Features:

• Roots:
  • Where the graph crosses the x-axis (y = 0)
  • Found by solving the quadratic equation
• y-intercept:
  • Found by substituting x = 0 → y = c
• Line of symmetry:
  • Vertical line that splits the graph evenly
  • Equation: x = –b/(2a)
• Turning point (not required for CORE) is the highest or lowest point — only its x-coordinate may be used via symmetry

Sketching Tips:

Linear:

1. Use two points (e.g., when x = 0 and x = 1)
2. Draw straight line through them

Quadratic:

1. Mark the y-intercept (x = 0, y = c)
2. Find the x-intercepts (solve ax² + bx + c = 0)
3. Draw symmetrical U or ∩ shape through those points

Quick Identification

Examples:

1. y = x + 2
   • Straight line
   • y-intercept = 2
   • Gradient = 1
2. y = x² – 4
   • Parabola (U-shape)
   • Roots at x = –2 and x = 2
   • y-intercept = –4
   • Line of symmetry at x = 0
3. y = –x² + 3x
   • Parabola (∩-shape)
   • Opens downward
   • y-intercept = 0
   • Roots: solve –x² + 3x = 0 → x(x – 3) = 0 → x = 0, 3

Common Exam Tasks

• Identify graphs from equations
• Match tables to sketches
• Sketch linear/quadratic graphs from key points
• Identify roots and y-intercepts
• Recognise line of symmetry from a quadratic

CORE-level focuses on basic recognition, plotting, symmetry, intercepts, and shape — turning point details are not required.