A12: Other Graphs
Recognise, sketch and interpret graphs of linear, quadratic, cubic, reciprocal, exponential and trigonometric functions
📋 Key Concepts
Types of graphs: Different equations produce different shaped graphs. Being able to recognise these shapes is essential.
| Type | Equation | Shape |
|---|---|---|
| Linear | y = mx + c | Straight line |
| Quadratic | y = ax² + bx + c | Parabola (U or ∩) |
| Cubic | y = ax³ + ... | S-curve |
| Reciprocal | y = a⁄x | Two curved branches |
| Exponential | y = ax or y = kax | Increasing/decreasing curve |
| Trigonometric | y = sin x, cos x, tan x | Wave patterns |
📝 Cubic Graphs
Cubic equation: y = ax³ + bx² + cx + d (highest power of x is 3)
Shape: An S-shaped curve. If a > 0, the curve goes from bottom-left to top-right. If a < 0, it goes from top-left to bottom-right.
Example 1
Sketch y = x³
Key points:
- Passes through origin (0, 0)
- Positive cubic: starts bottom-left, ends top-right
- As x → ∞, y → ∞; as x → -∞, y → -∞
Example 2
Sketch y = -x³
Key points:
- Passes through origin (0, 0)
- Negative cubic: starts top-left, ends bottom-right
📝 Reciprocal Graphs
Reciprocal equation: y = a⁄x or y = a⁄x²
Features:
- Two separate curved branches
- Never touches the axes (asymptotes)
- y = 1⁄x: branches in 1st and 3rd quadrants
- y = 1⁄x²: branches in 1st and 2nd quadrants
Example 3
Sketch y = 1⁄x
Key points:
- Asymptotes at x = 0 and y = 0
- Branch in quadrant 1 (positive x, positive y)
- Branch in quadrant 3 (negative x, negative y)
📝 Exponential Graphs
Exponential equation: y = ax where a > 0
Features:
- y = 2x: starts near x-axis, increases rapidly
- y = (½)x or y = 2-x: decreases towards x-axis
- All pass through (0, 1)
- Asymptote at y = 0 (x-axis)
Example 4
Sketch y = 2x
Key points:
- Passes through (0, 1)
- As x increases, y increases rapidly
- As x decreases, y approaches 0
📝 Trigonometric Graphs
Sine graph: y = sin x
- Wave between y = -1 and y = 1
- Passes through origin (0, 0)
- Period = 360°
Cosine graph: y = cos x
- Wave between y = -1 and y = 1
- Starts at maximum (0, 1)
- Period = 360°
Tangent graph: y = tan x
- Repeats every 180°
- Has vertical asymptotes at x = 90°, 270°, etc.
📝 Recognising Graphs
Example 5
Match these equations to their graphs:
a) y = x² - 3
b) y = x³
c) y = 1⁄x
d) y = 2x
Solutions:
a) Parabola (U-shape), y-intercept at -3
b) S-curve through origin
c) Two curved branches, asymptotes at axes
d) Curve increasing from (0, 1)
🧮 Worked Examples — Exam Style
Non-linear graphs are recognised by SHAPE. Learn the five families — cubic, reciprocal, exponential, trigonometric — and 'which graph is which' becomes free marks.
Describe the key features of the graph of y = x³.
Passes through the origin, S-shaped (positive cubic): flattens briefly at the origin, falls to the bottom-left, rises to the top-right.
B1 for S-shape through origin. Negative cubics (−x³) are reflected: top-left to bottom-right.
Sketch y = 2ˣ and label where it crosses the y-axis.
Starts near the x-axis on the left (never touches), rises increasingly steeply.
Crosses the y-axis at (0, 1) because 2⁰ = 1 — every exponential y = aˣ passes through (0, 1).
M1 for increasing-curve shape with asymptote at the x-axis, A1 for (0, 1).
Exam wording to expect: "Sketch", "match the equation to the graph", "write down the coordinates where the graph crosses …".
❓ Practice Questions
Q1: What shape is the graph of y = x³ - 2x?
Q2: Where are the asymptotes on y = 2⁄x?
Q3: What point does y = 3x pass through?
Q4: Sketch y = sin x for 0° ≤ x ≤ 360°. Where does it cross the x-axis?
Q5: Which graph has branches in quadrants 1 and 2: y = 1⁄x or y = 1⁄x²?
Q6: What is the maximum value of y = cos x?
Q7: Sketch y = cos x for 0° ≤ x ≤ 360°. Where does it cross the x-axis?
Q8: Which is steepest near x = 0: y = 2ˣ or y = x²?
✅ Answers
- S-shaped cubic curve
- x = 0 and y = 0 (the axes)
- (0, 1)
- x = 0°, 180°, 360°
- y = 1⁄x² (both branches are above x-axis)
- 1
🧮 Worked Examples — Harder Exam Style
Match each equation to its graph shape: (a) y = x³ (b) y = 1/x (c) y = 3ˣ (d) y = sin x — given graphs: S-curve through origin; two branches in opposite quadrants; curve through (0, 1) rising steeply; smooth wave between −1 and 1.
(a) S-curve through the origin → y = x³
(b) Two branches in opposite quadrants (1 and 3 for positive k) → y = 1/x
(c) Through (0, 1), rising steeply → y = 3ˣ
(d) Wave oscillating between −1 and 1 → y = sin x
B1 each. Elimination makes matching safe: exponential is the only one pinned at (0, 1); reciprocal is the only one with two separate branches.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. S-shaped cubic curve
Q2. y = 2/x has asymptotes x = 0 (the y-axis) and y = 0 (the x-axis) — the curve gets arbitrarily close but never touches
Q3. y = 3ˣ passes through (0, 1) since 3⁰ = 1
Q4. sin x crosses the x-axis at 0°, 180°, 360°, peaking at 90° (y = 1) and bottoming at 270° (y = −1)
Q5. y = 1/x² — squaring makes both branches positive, so both sit ABOVE the x-axis in quadrants 1 and 2
Q6. Maximum of cos x is 1 (at x = 0° and 360°)
Q7. cos x starts at (0°, 1), falls to (180°, −1), returns to (360°, 1); crosses the x-axis at 90° and 270°
Q8. y = 2ˣ — exponential growth climbs faster than any polynomial near and beyond small x; at x = 0 both equal 1, but 2ˣ pulls away immediately
📌 Key Facts to Memorise
- Cubic (x³): S-shape through the origin; negative cubic is upside-down
- Reciprocal (k/x): two branches; asymptotes are the axes; k/x² has both branches above
- Exponential (aˣ): passes (0, 1); horizontal asymptote y = 0; grows fast for a > 1
- sin x: 0 → 1 (90°) → 0 (180°) → −1 (270°) → 0 (360°); cos x starts at 1
- Trig graphs repeat every 360°; maximum 1, minimum −1
🗺️ In the Exam
Graph recognition: 1–2 marks on Paper 2/3. Trig graphs gate the Higher graph-transformation questions; reciprocals gate asymptote reasoning.
Links to: straight lines (A9), quadratic graphs (A11), transformations (A13).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Learn the shapes: linear (straight), quadratic (U), cubic (S), reciprocal (two branches)
- Reciprocal graphs never touch the axes
- Exponential graphs pass through (0, 1)
- Trig graphs repeat periodically
- Always label key points: intercepts, asymptotes
🧠 Problem-Solving Strategies
Problem-Solving
Identify graphs by shape: straight line (linear), U/∩ (quadratic), S-curve (cubic), two branches near axes (reciprocal), rapid growth/decay (exponential), waves (trig). Always look for key features: intercepts, asymptotes, turning points, and whether y increases or decreases.
Multi-Step Problem
A population of bacteria grows according to P = 100 × 2t, where t is time in hours. (a) Find P when t = 0, 1, 2, 3. (b) What type of graph is this? (c) When does the population exceed 10,000?
Solution:
(a) t=0: 100, t=1: 200, t=2: 400, t=3: 800
(b) Exponential growth graph (starts at 100, doubles each hour)
(c) 100 × 2t > 10000 → 2t > 100 → t > log₂(100) ≈ 6.64, so after about 7 hours
⚠️ Common Errors
Watch Out!
1. Wrong: A reciprocal graph touches the axes Correct: A reciprocal graph has asymptotes at the axes — it never touches them
2. Wrong: y = 2x passes through (1, 0) Correct: y = 2x passes through (0, 1) — when x = 0, y = 2⁰ = 1
3. Wrong: A cubic graph always passes through the origin Correct: Only y = x³ passes through the origin; y = x³ + 2 is shifted up and doesn't
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Match each equation to its graph description: (i) y = x³ - 3x, (ii) y = 4⁄x, (iii) y = 3x, (iv) y = sin x. A: Wave between -1 and 1, B: Two branches in opposite quadrants, C: S-curve crossing x-axis 3 times, D: Curve passing through (0,1) increasing rapidly.
(i) y = x³ - 3x → C (cubic, factorises to x(x²-3) = x(x-√3)(x+√3), 3 x-intercepts)
(ii) y = 4⁄x → B (reciprocal, branches in 1st and 3rd quadrants, asymptotes at axes)
(iii) y = 3x → D (exponential, passes through (0,1), increases rapidly)
(iv) y = sin x → A (trigonometric wave, oscillates between -1 and 1)
Mark scheme: 1.5 marks each for correct matching with reasoning (6 total).
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Drug concentration in blood is modelled by C = 10 × (0.5)t mg/L, where t is hours after injection.
(a) What is the initial concentration?
(b) What type of graph does this produce?
(c) The drug is effective above 1 mg/L. For how many hours is it effective?
Answers: (a) C = 10 × 0.5⁰ = 10 mg/L. (b) Exponential decay — decreasing towards 0. (c) 10 × 0.5t > 1 → 0.5t > 0.1 → t < log0.5(0.1) ≈ 3.32 hours.
📝 Exam Questions by Topic
🎬 Video Resources
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