Sketching Curves

2.11 Sketching Curves – Cheat Sheet

1. Linear Functions

• Form: ax + by = c or y = mx + c
• Graph: Straight line
• m = gradient, c = y-intercept
• Gradient: rise/run = change in y ÷ change in x
• Intercepts:
  • y-intercept: set x = 0
  • x-intercept: set y = 0

Example: y = 2x + 3 → Gradient = 2, intercept (0, 3)

2. Quadratic Functions

• Form: y = ax² + bx + c
• Graph: U-shaped parabola (a > 0 opens up, a < 0 opens down)
• Roots: Solve y = 0
• Turning Point: Complete the square
  • y = (x + p)² + q → turning point (−p, q)
• Axis of Symmetry: x = −b / (2a)

Example: y = x² − 4x + 3

• Factorise: (x − 1)(x − 3) → roots: x = 1, 3
• Axis of symmetry: x = 2
• Turning point: (2, −1)

3. Cubic Functions

• Form: y = ax³ + bx² + cx + d
• Graph: S-shape (can have 1 or 2 turning points)
• Intercepts:
  • y-intercept: set x = 0
  • x-intercepts: solve y = 0 (factorisation may be needed)

Example: y = x³ − x

• Factorise: x(x − 1)(x + 1) → roots: x = 0, 1, −1

4. Reciprocal Functions

• Form: y = a/x or y = a/(x − h) + k
• Asymptotes:
  • Vertical: x = 0 (or x = h if shifted)
  • Horizontal: y = 0 (or y = k if shifted)
• Shape: Two separate curves in opposite quadrants (hyperbola)

Example: y = 1/x

• Vertical asymptote: x = 0
• Horizontal asymptote: y = 0

5. Exponential Functions

• Form: y = arˣ + b, r > 0
• Growth: r > 1 → increases rapidly
• Decay: 0 < r < 1 → decreases towards horizontal asymptote
• Horizontal asymptote: y = b

Example: y = 2ˣ

• Passes through (0, 1), doubles each step, asymptote y = 0

6. Key Features to Recognise and Sketch

7. Steps to Sketching Any Curve

1. Identify function type.
2. Find y-intercept (x = 0).
3. Find x-intercept(s) (y = 0).
4. Find turning point(s) if needed.
5. Determine symmetry and asymptotes.
6. Plot key points and sketch smoothly.

8. Examples Table for Quick Reference