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Sketching curves

Asymptotes – IGCSE Summary Notes
IGCSE MATHS · SUMMARY NOTES

Understanding Asymptotes

Everything IGCSE candidates need to know — definitions, types, rules, and exam tips.

Definition

An asymptote is a line that a curve approaches but never actually reaches or crosses. As the x or y values get very large (or very small), the curve gets closer and closer to the asymptote — but the gap never quite closes to zero.

The Three Types

↕

Vertical Asymptote

Most Common at IGCSE

A vertical line the curve approaches from the left or right. The curve shoots up to ±∞ as x gets close to this value.

Form: x = a

How to find it: Set the denominator equal to zero and solve for x.

y = 1/x → x = 0 y = 1/(x−3) → x = 3 y = 2/(x+5) → x = −5
↔

Horizontal Asymptote

Key for Reciprocals & Exponentials

A horizontal line the curve approaches as x → +∞ or x → −∞. The curve flattens out and gets closer to this y-value.

Form: y = b

How to find it: Consider what happens to y as x becomes very large.

y = 1/x → y = 0 y = 2ˣ → y = 0 y = 3 + 1/x → y = 3
↗

Oblique Asymptote

Beyond Core IGCSE

A diagonal (slanted) line the curve approaches as x → ±∞. Occurs when degree of numerator is exactly one more than denominator.

Form: y = mx + c

Note: This is rarely examined at IGCSE but good to be aware of for extended/higher tier papers.

Asymptotes by Function Type

Function Vertical Asymptote Horizontal Asymptote Type
y = k/xx = 0y = 0Both
y = k/(x − a)x = ay = 0Both
y = b + k/xx = 0y = bBoth
y = aˣNoney = 0Horizontal
y = aˣ + cNoney = cHorizontal
y = ax² + bx + cNoneNoneNo asymptotes

Key Rules to Remember

01

Never Touching

The curve always approaches the asymptote but never crosses or touches it.

02

Denominator = 0

Vertical asymptotes occur where denominator = 0 and numerator ≠ 0.

03

Two Branches

y = k/x produces two separate branches, one in each pair of opposite quadrants.

04

Label on Sketches

Draw asymptotes as dashed lines and label them with their equation.

Exam Tips

✏️

Always draw asymptotes as dashed lines. A solid line suggests the curve actually reaches there.

🔍

Check for shifts. In y = 1/(x−2) + 3, asymptotes shift to x = 2 and y = 3.

📐

Reciprocal symmetry. y = k/x is symmetrical about y = x and y = −x.

⚠️

Polynomials have NO asymptotes. Quadratics and cubics don't approach a fixed line.

The Golden Rule

An asymptote is a boundary the curve respects but never breaks. Draw it as a dashed line, label it — and your sketch is already ahead of most answers.

IGCSE Mathematics · Asymptotes · Summary Notes
Intercepts quick guide · curve sketching

Quick intercepts for curve sketching

Two simple rules: y-intercept (set x=0), x-intercept (set y=0).

y‑intercept

RULE: set x = 0

Where the curve crosses the y‑axis. Always one point (or none if undefined).

y = (x+2)/(x-3)
set x = 0 → y = (0+2)/(0-3) = 2/(-3) = -2/3
➔ y‑intercept: (0, -2/3)
y = 2x + 1
set x = 0 → 20+1 = 1+1 = 2
➔ (0, 2)
y = x² - 4
x=0 → y = -4 → (0, -4)

x‑intercept(s)

RULE: set y = 0

Solve equation for x. May be zero, one, or multiple points.

y = (x+2)/(x-3) = 0
numerator = 0 → x+2=0 → x = -2
(denominator ignored for roots) ➔ (-2, 0)
y = 2x - 8
set 2x - 8 = 0 → 2x = 8 → x = 3
➔ (3, 0)
y = x² - 5x + 6
factor: (x-2)(x-3)=0 → x = 2, 3
➔ (2,0) and (3,0)

Special situations – no intercepts

No y‑intercept

If function undefined at x = 0.
e.g. y = 1/x → x=0 gives division by zero → no y‑intercept.

No x‑intercept

If equation y=0 has no (real) solution.
e.g. y = x²+4 → x² = -4 impossible → no x‑intercept.

Intercepts at a glance

Function typey‑intercept (x=0)x‑intercept(s) (y=0)
Linear: y = mx + cy = cset mx+c=0 → x = -c/m
Quadratic: y = ax²+bx+cy = csolve ax²+bx+c=0 (factor / formula)
Rational: y = p(x)/q(x)p(0)/q(0) (if q(0)≠0)solve p(x)=0 (ignore denominator)
Exponential: y = ax + k1 + k (since a⁰=1)set ax = -k → only if -k>0

Step‑by‑step: intercepts from equations

y = (x² - 9) / (x + 2)
🔹 y‑int: set x=0 → (0-9)/(0+2) = -9/2 = -4.5 → (0, -4.5)
🔹 x‑int: set numerator = 0 → x² - 9 = 0 → (x-3)(x+3)=0 → x = 3, x = -3 → (3,0) and (-3,0).
note: denominator ≠0 at these points ✓
y = (x+5) / (x² - 1)
🔹 y‑int: x=0 → 5/(-1) = -5 → (0, -5)
🔹 x‑int: numerator=0 → x+5=0 → x = -5 → (-5,0). (denominator at x=-5 ≠ 0, fine)
y = 3·2x - 12
🔹 y‑int: x=0 → 3·1 -12 = -9 → (0, -9)
🔹 x‑int: set 3·2x -12 = 0 → 2x = 4 → x = 2 → (2,0)
Common mistake: For rational functions, don't set denominator = 0 when looking for x‑intercepts. Only numerator matters for x‑intercepts (points where y=0). Denominator = 0 gives asymptotes, not intercepts.
10‑second summary:
x=0 → y‑int   |   y=0 → x‑int
Curve Sketching – IGCSE Guide

IGCSE Mathematics

Curve
Sketching

A complete step-by-step guide for IGCSE students

Curve sketching is one of the most visual skills in IGCSE Maths. Follow these eight steps every time and you'll never miss a key feature again.

The Eight Steps

1

Identify the Type of Function

Recognise what kind of curve you're dealing with before doing anything else.

  • Linear y = mx + c → straight line
  • Quadratic y = ax² + bx + c → parabola
  • Cubic y = ax³ + … → S-shaped curve
  • Reciprocal y = k/x → hyperbola with asymptotes
  • Exponential y = aˣ → rapid growth or decay
2

Find the Y-intercept

Set x = 0 and solve for y. This gives you the point where the curve crosses the vertical axis.

3

Find the X-intercepts (Roots)

Set y = 0 and solve for x.

  • For quadratics: factorise, use the quadratic formula, or complete the square
  • The number of roots tells you how many times the curve crosses the x-axis
4

Find the Vertex or Turning Point

For quadratics y = ax² + bx + c:

  • Use x = −b ÷ 2a to find the x-coordinate
  • Substitute back to find the y-coordinate
  • If a > 0 → U-shaped (minimum point)
  • If a < 0 → ∩-shaped (maximum point)
5

Check for Symmetry

Quadratics are symmetrical about the vertical line through their turning point. Reciprocal graphs are symmetrical about y = x and y = −x.

6

Identify Asymptotes

  • For y = k/x: asymptotes are x = 0 and y = 0 (the axes)
  • For y = aˣ: asymptote is y = 0 (the x-axis)
Remember: the curve gets close to an asymptote but never actually touches it.
7

Consider the Behaviour at Extremes

Think about what happens as x → very large or very small (very negative). Does the curve rise or fall? This helps you draw the "tails" of the curve correctly.

8

Plot Key Points & Sketch

Mark all intercepts and turning points on your axes, then join them with a smooth curve — not straight lines between dots! Make sure the shape matches the function type you identified in Step 1.

Quick Reference Table

Function Shape Key Features
y = ax² + bx + c Parabola 1 turning point, line of symmetry
y = ax³ Cubic Passes through origin, S-shape
y = k/x Hyperbola Two branches, two asymptotes
y = aˣ Exponential Always positive, y-intercept at (0, 1)

✏️ Top Tips for the Exam

  • Always label intercepts, turning points, and asymptotes with their coordinates.
  • Use a ruler for axes only — the curve itself should be freehand and smooth.
  • Double-check your intercepts by substituting back into the equation.

IGCSE Mathematics  ·  Curve Sketching  ·  Study Guide

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