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S4 Maths: Transformations of Graphs (Paper 4 (error clinic))
S4 Maths: Transformations of Graphs (Paper 4 (error clinic))
Paper 4 (error clinic): ten graph-transformation questions with answers and one-line notes.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S4 Maths: Transformations of Graphs |
| Level | S4 |
02 Questions (10 items)
- FoundationShift y = f(x) 4 units down. Write the new equation and the image of (2, 9).
- FoundationShift y = f(x) 5 units right. Write the new equation and the image of (3, 1).
- FoundationReflect y = f(x) in the x-axis and track (-2, 6), then reflect the result in the y-axis. Give both images.
- IntermediateCompress y = f(x) vertically by a factor of 2. Write the new equation and the image of (4, 10).
- IntermediateCompress y = f(x) horizontally by a factor of 4. Write the new equation and the image of (8, 3).
- IntermediateGive the vertex and opening direction of y = (x – 1) squared plus 2, and describe the transformations from y = x squared.
- ChallengeReflect y = x squared in the x-axis and then shift it 3 units up. Write the new equation and the vertex.
- ChallengeCompare y = f(x) + 3 with y = f(x + 3): how does each graph move?
- ChallengeShift y = f(x) 3 units up and then compress it horizontally by a factor of 2. Find the image of (2, 4).
- Error redoA student claims that y = f(2x) shifts the graph 2 units horizontally. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | y = f(x) – 4; the point moves to (2, 5) | Subtracting 4 outside lowers the y coordinate from 9 to 5 while x stays the same. |
| 2 | y = f(x – 5); the point moves to (8, 1) | A rightward shift replaces x by x minus 5, so the x coordinate goes from 3 to 8 and y is unchanged. |
| 3 | First (-2, -6), then (2, -6) | Reflecting in the x-axis changes only the sign of y and reflecting in the y-axis changes only the sign of x, so together they act like a reflection in the origin. |
| 4 | y = f(x)/2; the point moves to (4, 5) | Multiplying outside by one half turns the y value from 10 into 5 while x stays the same. |
| 5 | y = f(4x); the point moves to (2, 3) | Solving 4x = 8 gives x = 2 while the y value stays at 3. |
| 6 | Vertex (1, 2) opening upwards: 1 unit right and then 2 units up | The minus 1 inside shifts the graph 1 unit right, the plus 2 outside shifts it 2 units up, and a positive squared coefficient opens upwards. |
| 7 | y = minus x squared + 3, vertex (0, 3), opening downwards | The reflection makes the graph open downwards and the upward shift of 3 lifts the vertex from (0, 0) to (0, 3). |
| 8 | y = f(x) + 3 moves 3 units up; y = f(x + 3) moves 3 units left | The change outside the function is vertical while the change inside is horizontal, and the two move in opposite senses. |
| 9 | The point moves to (1, 7) | Shifting up by 3 gives (2, 7) and the horizontal compression then halves the x coordinate to 1. |
| 10 | It mistakes a stretch for a translation: f(2x) compresses the graph horizontally by 2, making it narrower without moving it sideways. | The coefficient rescales x instead of shifting it, since only adding or subtracting a constant translates a graph, while multiplying by 2 halves every x coordinate. |
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.
For any graph transformation, first ask whether the change sits outside the function or inside it. Changes outside (adding k, multiplying by a, adding a minus sign) move the graph up or down, stretch it vertically or reflect it in the x-axis. Changes inside the variable (x plus h, x multiplied by k) move or stretch the graph horizontally, and the direction is the opposite of what most students expect: f(x + 3) shifts the graph 3 units to the left, not to the right. To track a single point, put its x into the inside change, feed the result into the outside change, and work out the new coordinates step by step instead of guessing. For two-step transformations, do the inside first and the outside second, because a different order gives a different image. Finish by checking that the vertex moved and the opening direction flipped correctly, then substitute a convenient value such as x = 0 to confirm.Tutor Registration →
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.
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