IGCSE Mathematics 0580 – Algebra: Graphs of Functions
This set of notes covers the five families of functions required for the Cambridge IGCSE Mathematics (0580) syllabus (2025‑2027). Core material is essential for all candidates; sections marked Extended are useful for higher‑level work (E‑questions).
1. Linear Functions (Core)
- General form (syllabus requirement):
y = ax + b, ≠ 0
- Vertical line (Core – also required):
x = k, k is a constant
- Domain: ℝ (all real numbers)
- Range: ℝ
- Intercepts
- y‑intercept (0, )
- x‑intercept (−b/a, 0) provided ≠ 0
- Gradient
- a > 0 – rising line
- a < 0 – falling line
- a = 0 – horizontal line y =
- Transformations (Core)
- Vertical shift
y = ax + b + k (up by k if k > 0)
- Horizontal shift
y = a(x – h) + b (right by h if h > 0)
- Reflection in the x‑axis
y = –ax – b
- Reflection in the y‑axis
y = –ax + b
Worked example – Sketching a linear function
Sketch y = 2x – 3.
- Gradient = 2 → rising line.
- y‑intercept (0, −3).
- x‑intercept x = 3/2 → (1.5, 0).
- Plot the two intercepts and draw the straight line through them.
2. Quadratic Functions (Core)
- Standard form (required):
y = ax² + bx + c, ≠ 0
- Vertex (completed‑square) form – useful for sketching:
y = a(x – h)² + k, vertex ( , )
- Factored form (Core – for finding roots quickly):
y = a(x – r₁)(x – r₂)
- Domain: ℝ
- Range
- Axis of symmetry:
x = h or x = –b/(2a)
- Intercepts
- y‑intercept (0, c)
- x‑intercepts solve
ax² + bx + c = 0 using the quadratic formula
x = [‑b ± √(b² – 4ac)] / (2a)
- Transformations (Core)
- Vertical stretch/compression multiply the whole expression by
- Horizontal shift replace x by (x – h)
- Vertical shift add k to the expression
- Reflection in the x‑axis change the sign of
Worked example – Using the factored form
Sketch y = –(x – 1)(x + 2).
- Roots are x = 1 and x = −2 → points (1, 0) and (−2, 0).
- Since = −1 < 0, the parabola opens downwards.
- Axis of symmetry x = (1 + (−2))/2 = −0.5.
- Vertex plug x = −0.5 into the equation → y = –(−0.5 – 1)(−0.5 + 2) = –(−1.5)(1.5) = 2.25.
Vertex (−0.5, 2.25).
- Plot the two roots, the vertex, and draw a smooth parabola.
3. Cubic Functions (Core + Extended)
- General form (required):
y = ax³ + bx² + cx + d, ≠ 0
- Domain: ℝ
- Range: ℝ
- End behaviour (determined by )
- a > 0 → as x → −∞, y → −∞ and as x → +∞, y → +∞
- a < 0 → as x → −∞, y → +∞ and as x → +∞, y → −∞
- Turning points (Core)
A cubic may have up to two turning points (a maximum and a minimum). For Core candidates you only need to recognise the possible “S‑shaped” pattern; exact coordinates are not required.
- Inflection point (Extended only)
For the reduced cubic y = ax³ + cx + d the point of inflection is at x = –b/(3a). This is useful for extended questions (E2.12).
- Intercepts
- y‑intercept (0, d)
- x‑intercepts real roots of the cubic equation (1 or 3 real roots). Factoring by inspection or using the factor theorem is the usual exam technique.
- Transformations (Core)
- Vertical stretch/compression multiply by
- Horizontal shift replace x by (x – h)
- Vertical shift add k
- Reflection in the x‑axis change the sign of
Worked example – Sketching a cubic (Core)
Sketch y = (x – 1)(x + 1)(x – 2).
- Expand (optional) →
y = x³ – 2x² – x + 2 so = 1 > 0.
- Roots: x = 1, −1, 2 → points (1,0), (−1,0), (2,0).
- End behaviour: rises to +∞ on the right, falls to –∞ on the left.
- Plot the three intercepts, draw a smooth curve that starts low on the left, passes through (−1,0), rises to a maximum between (−1,0) and (1,0), falls through (1,0), reaches a minimum between (1,0) and (2,0), then rises through (2,0) and continues upwards.
Worked example – Inflection point (Extended)
Find the inflection point of y = 2x³ – 6x + 4.
- Identify = 2, b = 0 (there is no x² term).
- Inflection x‑coordinate x = –b/(3a) = 0.
- y‑coordinate y = 2·0³ – 6·0 + 4 = 4.
- Inflection point (0, 4).
4. Reciprocal Functions (Core + Extended)
- Basic form (Core requirement):
y = k / x, k ≠ 0
- General (Extended) form:
y = k / (x – h) + c
- Domain: x ≠ 0 (Core) or x ≠ h (Extended)
- Range: y ≠ 0 (Core) or y ≠ c (Extended)
- Asymptotes
- Vertical x = 0 (Core) or x = h (Extended)
- Horizontal y = 0 (Core) or y = c (Extended)
- Intercepts
- y‑intercept set x = 0 → y = k/(‑h) + c (only when h ≠ 0)
- x‑intercept set y = 0 → x = h – k/c (only when c ≠ 0)
- Shape
- k > 0 → branches in quadrants I & III relative to the asymptotes.
- k < 0 → branches in quadrants II & IV.
- Transformations
- Horizontal shift h moves the vertical asymptote to x = h.
- Vertical shift c moves the horizontal asymptote to y = c.
- Vertical stretch/compression |k| changes how close the curve gets to the axes.
- Reflection in the x‑axis change the sign of k.
Worked example – Shifts and asymptotes (Extended)
Sketch y = –3 / (x – 2) + 1.
- Vertical asymptote x = 2.
- Horizontal asymptote y = 1.
- k = –3 < 0 → branches in quadrants II & IV relative to the asymptotes.
- Find intercepts:
- y‑intercept: x = 0 → y = –3/(0‑2) + 1 = –3/(‑2) + 1 = 1.5 + 1 = 2.5 → (0, 2.5).
- x‑intercept: set y = 0 → 0 = –3/(x‑2) + 1 → –1 = –3/(x‑2) → 1 = 3/(x‑2) → x‑2 = 3 → x = 5 → (5, 0).
- Plot the asymptotes as dashed lines, mark the intercepts, and draw the two hyperbolic branches respecting the asymptotes.
5. Exponential Functions (Core + Extended)
- Core form (required):
y = a·bˣ, a ≠ 0, b > 0, b ≠ 1
- Extended form (vertical shift):
y = a·bˣ + c
- Domain: ℝ
- Range
- If a > 0 → y > c (or y > 0 when c = 0)
- If a < 0 → y < c
- Horizontal asymptote
- Core form y = 0
- Extended form y = c
- Growth vs. decay
- Growth when b > 1
- Decay when 0 < b < 1
- Intercepts
- y‑intercept x = 0 → y = a + c
- x‑intercept (if it exists) solve a·bˣ + c = 0 →
x = log₍b₎(−c⁄a) provided −c⁄a > 0
- Transformations (Core)
- Vertical stretch/compression multiply by a.
- Horizontal stretch/compression replace x by kx ( equivalent to changing the base to b^{1/k}).
- Reflection in the x‑axis change the sign of a.
- Vertical shift add c (only in the extended form).
Worked example – Exponential with a vertical shift (Extended)
Sketch y = 2·(½)ˣ – 3.
- Base b = ½ → decay (curve falls as x increases).
- a = 2 > 0, c = ‑3 → horizontal asymptote y = ‑3.
- y‑intercept: x = 0 → y = 2·1 – 3 = ‑1 → (0, ‑1).
- x‑intercept: solve 2·(½)ˣ – 3 = 0 → (½)ˣ = 3/2 → take log base ½:
x = log₍½₎(3/2) = –log₂(3/2) ≈ –0.585 → (‑0.585, 0).
- Plot the asymptote y = ‑3 as a dashed line, mark the intercepts, and draw the curve approaching y = ‑3 from above as x → +∞ and rising steeply as x → ‑∞.
6. Extended‑Only Topic: Surds & Rationalising Denominators
Although not part of the core graphing syllabus, simplifying surds is required for many extended questions (E1.18).
- Basic surd rules
- √(ab) = √a·√b (when a, b ≥ 0)
- √a / √b = √(a⁄b)
- (√a)² = a
- Rationalising a denominator
- Single surd:
1 / √a = √a / a
- Binomial of the form a ± √b: multiply numerator and denominator by the conjugate a ∓ √b.
Comparison of the Five Function Families
| Family |
General equation (Core) |
Domain |
Range |
Key asymptote(s) |
Typical shape |
| Linear |
y = ax + b or x = k |
ℝ |
ℝ |
None |
Straight line (incl. vertical line) |
| Quadratic |
y = ax² + bx + c |
ℝ |
y ≥ k (a > 0) or y ≤ k (a < 0) |
None |
Parabola (opens up or down) |
| Cubic |
y = ax³ + bx² + cx + d |
ℝ |
ℝ |
None |
S‑shaped curve (up to two turning points, one inflection point in extended work) |
| Reciprocal |
y = k / x (Extended: y = k/(x‑h) + c) |
x ≠ 0 (Extended: x ≠ h) |
y ≠ 0 (Extended: y ≠ c) |
Vertical x = 0 and Horizontal y = 0 (Extended: x = h, y = c) |
Two hyperbolic branches |
| Exponential |
y = a·bˣ (Extended: y = a·bˣ + c) |
ℝ |
y > c (a > 0) or y < c (a < 0) |
Horizontal y = 0 (Extended: y = c) |
Rapid increase (b > 1) or decrease (0 < b < 1) |