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S4 Maths: Transformations of Graphs (Paper 2 (intermediate))
S4 Maths: Transformations of Graphs (Paper 2 (intermediate))
Paper 2 (intermediate): 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)
- FoundationThe graph of y = f(x) is shifted 2 units to the left. Write the new equation and the image of (4, 6).
- FoundationStretch y = f(x) vertically by a factor of 3, then shift it 1 unit up; given f(2) = 4, find the image of (2, 4).
- FoundationCompress y = f(x) horizontally by a factor of 2. Write the new equation and the image of (6, 5).
- IntermediateReflect y = f(x) in the x-axis and then shift it 5 units up; given f(1) = 3, find the image of (1, 3).
- IntermediateShift y = f(x) one unit right and then two units up. Write the new equation and the image of (0, 4).
- IntermediateCompress y = f(x) horizontally by 2, then stretch it vertically by 2. Write the new equation and follow the point (5, 7) through both steps.
- ChallengeThe graph of y = x squared becomes y = (x + 3) squared minus 4. Give the new vertex and describe both transformations.
- ChallengeReflect y = f(x) in the y-axis and then shift it 2 units up. Write the new equation and the image of (2, 5).
- ChallengeA curve is reflected in the x-axis twice. How does the final graph compare with the original? Explain.
- Error redoA student claims that y = 2f(x) compresses the graph horizontally by half. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | y = f(x + 2); the point moves to (2, 6) | A shift of h units to the left replaces x by x plus h, so the image has x equal to 4 minus 2, which is 2, and y stays 6. |
| 2 | y = 3f(x) + 1; the point moves to (2, 13) | Multiplying by 3 gives 12 and adding 1 gives 13, so the image point has y coordinate 13. |
| 3 | y = f(2x); the point moves to (3, 5) | A horizontal compression by a factor of k replaces x by kx, and solving 2x = 6 gives x = 3 while y stays 5. |
| 4 | y = -f(x) + 5; the point moves to (1, 2) | Reflecting first gives y = minus 3, and adding 5 gives 2, so the image point is (1, 2). |
| 5 | y = f(x – 1) + 2; the point moves to (1, 6) | The x coordinate rises from 0 to 1 and the y coordinate from 4 to 6, since both steps are pure translations that keep the shape. |
| 6 | y = 2f(2x); (5, 7) goes to (2.5, 7) and then to (2.5, 14) | The horizontal compression divides x by 2 and the vertical stretch multiplies y by 2, so the inside and outside changes are handled in two separate steps. |
| 7 | Vertex (-3, -4): 3 units left and then 4 units down | Adding 3 inside shifts the graph 3 units left and subtracting 4 outside shifts it 4 units down, so the vertex moves from (0, 0) to (-3, -4). |
| 8 | y = f(-x) + 2; the point moves to (-2, 7) | The reflection changes only the sign of x, giving (-2, 5), and the upward shift of 2 then gives (-2, 7). |
| 9 | It coincides with the original graph y = f(x) | The first reflection changes the sign of every y value and the second changes it back, because two minus signs multiply to a plus. |
| 10 | The direction is wrong: multiplying outside by 2 stretches the graph vertically by 2, while a horizontal compression needs f(2x). | A factor outside the function changes only the y values, so the graph becomes taller rather than narrower; a horizontal compression requires replacing x by 2x inside. |
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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