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S4 Maths functions and graphs paper 4: ten common errors taken apart
S4 Maths functions and graphs paper 4: ten common errors taken apart
Ten items target the slips seen most often earlier: misplaced minus signs, lost middle terms, reversed composites, domain conditions and inverse values.
Direct answers Items 01 to 03 rebuild squares and signs, a denominator domain and bracket expansion, 04 to 06 the axis sign, composite order and a root domain, 07 to 09 the discriminant, a maximum and a shift, and 10 the habit of substituting instead of inverting.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 25 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S4 Maths Functions and Graphs |
| Level | S4 |
02 Questions (10 items)
- FoundationError drill: f(x) = 2x^2 – 3. Find f(-2); a student answered -11. What is correct?
- FoundationError drill: state the domain of f(x) = 1/(x – 2). A student wrote x not equal to 0. What is correct?
- FoundationError drill: expand (x + 5)^2. A student wrote x^2 + 25. What is correct?
- IntermediateError drill: find the axis of symmetry of y = x^2 + 8x + 3. A student answered x = 4. What is correct?
- IntermediateError drill: f(x) = x + 2 and g(x) = 3x. Find f(g(x)); a student wrote 3x + 6. What is correct?
- IntermediateError drill: state the domain of f(x) = sqrt(x – 4). A student wrote x >= 0 only. What is correct?
- ChallengeError drill: how many points does y = x^2 – 6x + 9 share with the x-axis? A student said 2. What is correct?
- ChallengeError drill: what is the maximum of y = -(x – 2)^2 + 7? A student said minimum 7. What is correct?
- ChallengeError drill: how does y = f(x) move to become y = f(x) + 3? A student said 3 units left. What is correct?
- Error redoError drill: f(x) = 4x + 1. Find the inverse value at 5. A student substituted 5 into f and got 21. What is correct?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 5 | Square first: (-2)^2 = 4, then 2 x 4 – 3 = 5. The -11 comes from keeping the minus outside too early. |
| 2 | x not equal to 2 | The restriction comes from the denominator x – 2, which may not be zero, so x is not 2; x = 0 is allowed. |
| 3 | x^2 + 10x + 25 | Squaring means multiplying the whole bracket by itself, so the middle term 2 x x x 5 = 10x must appear. |
| 4 | x = -4 | The axis is x = -b / (2a) with b = 8 and a = 1, giving x = -4; dropping the minus gives 4. |
| 5 | 3x + 2 | In f(g(x)) the whole of 3x goes into f, giving 3x + 2; 3x + 6 comes from reversing the order. |
| 6 | x >= 4 | The expression under the root is x – 4 and it must not be negative, so x >= 4; the root sign alone is not the condition. |
| 7 | One point (discriminant = 0) | The discriminant is (-6)^2 – 4 x 1 x 9 = 0; zero means a single contact point, here (-3, 0). |
| 8 | Maximum 7, reached at x = 2 | The coefficient of the squared term is negative so the curve opens downward and has a maximum, at the point where the squared term is zero. |
| 9 | A shift of 3 units upward | Adding outside the function changes only the y values, so the graph rises; only an addition inside the bracket moves it sideways. |
| 10 | 1 | An inverse works backwards: 5 = 4x + 1 gives x = 1, while substituting 5 computes f(5) instead. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
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