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S4 Maths: Transformations of Graphs (Paper 3 (challenge))
S4 Maths: Transformations of Graphs (Paper 3 (challenge))
Paper 3 (challenge): 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)
- FoundationCompress y = f(x) horizontally by a factor of 3. Write the new equation and the image of (12, 5).
- FoundationStretch y = f(x) horizontally by a factor of 2. Write the new equation and the image of (3, 5).
- FoundationThe graph of y = af(x) passes through (2, 9) and f(2) = 3. Find a.
- IntermediateThe graph of y = f(x) + k passes through (5, 2) and f(5) = -7. Find k.
- IntermediateA transformation moves (7, 4) to (4, 4) using a horizontal shift only. Write the new equation.
- IntermediateGiven f(1) = 4, stretch the graph vertically by 2, reflect it in the x-axis and then shift it 5 units up. Find the image of (1, 4).
- ChallengeFor y = x squared and y = (2x) squared, find the x values that give y = 16, and describe the relation between the two graphs.
- ChallengeDescribe how y = 3f(x) and y = f(2x) each change the shape of a graph and its enclosed area.
- ChallengeShift y = f(x) two units left and then reflect it in the y-axis. Write the resulting equation.
- Error redoA student claims that y = f(-x) and y = -f(x) describe the same transformation. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | y = f(3x); the point moves to (4, 5) | Solving 3x = 12 gives x = 4 while the y value stays at 5. |
| 2 | y = f(x/2); the point moves to (6, 5) | A horizontal stretch by 2 replaces x by x over 2, so solving x over 2 equals 3 gives x = 6. |
| 3 | a = 3 | Substituting the point gives 9 equals a times 3, so a is 3 and the graph is stretched vertically by a factor of 3. |
| 4 | k = 9 | Substituting the point gives 2 equals minus 7 plus k, so k is 9 and the graph is shifted 9 units up. |
| 5 | y = f(x + 3) | The x coordinate drops from 7 to 4, a shift of 3 to the left, so x is replaced by x plus 3. |
| 6 | The point moves to (1, -3) | The stretch gives 8, the reflection gives minus 8 and adding 5 gives minus 3, while x is unchanged throughout. |
| 7 | x is 4 or -4 for the first graph and 2 or -2 for the second, which is the first compressed horizontally by a factor of 2 | Solving x squared equals 16 gives plus or minus 4, and solving (2x) squared equals 16 gives plus or minus 2, so the graph is squeezed horizontally without changing height. |
| 8 | y = 3f(x) triples the height so the area triples; y = f(2x) halves the width so the area halves | Multiplying outside by 3 triples the height and hence the area, while replacing x by 2x halves the width and hence the area. |
| 9 | y = f(-x + 2) | The shift gives f(x + 2) and replacing x by minus x then gives f(-x + 2); the order of the two steps matters. |
| 10 | The axis is wrong: f(-x) reflects in the y-axis while -f(x) reflects in the x-axis, so they are different. | The first changes the sign of the input, flipping the graph left to right, while the second changes the sign of the output, flipping it up to down. |
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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