Year 11 Maths Advanced Module 1 ⏱ ~45 min Lesson 12 of 15

Combined Transformations

In a video game, every character on screen is just a basic image that has been stretched, flipped, rotated, and moved into position. Game developers do not redraw the character for every frame — they apply combined transformations. In this lesson, you will learn to do the same thing with functions: stack multiple transformations together and read the result like a pro.

🎮
Printable worksheet

Download this lesson's worksheet

Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.

Download PDF Open printable version
Think First

The graph of $y = x^2$ has its vertex at $(0, 0)$. How would you transform this graph so that it opens downward, is twice as steep, and has its vertex at $(3, -2)$? Try to write an equation that achieves all three changes at once.

Type your initial response below — you will revisit this at the end of the lesson.

Write your initial response in your book. You will revisit it at the end of the lesson.

Write your initial thinking in your book
Saved
📐

Formula Reference — This Lesson

$y = af(b(x - h)) + k$
$a$ = vertical dilation (& reflection if $a < 0$) $b$ = horizontal dilation (& reflection if $b < 0$) $h$ = horizontal translation $k$ = vertical translation
Key insight: Always identify transformations from the "inside out": horizontal translation → horizontal dilation → vertical dilation → vertical translation.
📖 Know

Key Facts

  • How to combine translations, reflections, and dilations in one equation
  • The standard form $y = af(b(x - h)) + k$
  • How each parameter affects the graph
💡 Understand

Concepts

  • Why the order of reading transformations matters
  • How to find the image of a point after multiple transformations
  • Why some transformations commute and others do not
✅ Can Do

Skills

  • Write the equation of a graph after multiple transformations
  • Describe the transformations from a given equation
  • Find the new coordinates of key points after combined transformations
  • Sketch graphs with combined transformations

Misconceptions to Fix

✗

Wrong: The domain of a function is always all real numbers.

✗

Right: The domain depends on the function; rational functions exclude values that make the denominator zero.

📚 Core Content

Key Terms
FunctionA relation where each input has exactly one output.
DomainThe set of all possible input values for a function.
RangeThe set of all possible output values for a function.
Inverse FunctionA function that reverses the effect of the original function.
QuadraticA polynomial of degree 2, in the form ax² + bx + c.
DiscriminantThe expression b² - 4ac that determines the nature of quadratic roots.
🔢

Building the General Form

All the transformations you have learned so far can be combined into a single, powerful equation:

$$y = af(b(x - h)) + k$$

Each letter in this equation controls a specific transformation:

How video game engines position characters. A game character starts as a basic model in a local coordinate system. To place it on screen, the engine applies a scale (dilation), possibly flips it (reflection), and then moves it to the correct position (translation). The final on-screen coordinates are computed using exactly the same transformation rules you are learning now.

Reading Transformations from the Equation

The safest way to read combined transformations is from the inside out:

  1. Start with the innermost change: $x - h$ tells you the horizontal translation
  2. Next, $b(x - h)$ tells you the horizontal dilation (and any $y$-axis reflection)
  3. Then, $af(\dots)$ tells you the vertical dilation (and any $x$-axis reflection)
  4. Finally, $+ k$ tells you the vertical translation

Order Matters (Sometimes)

Some transformations can be applied in any order without changing the final result:

However, translations and dilations along the same axis generally do not commute. That is why we write the equation in the standard form above — the parentheses fix the correct order.

⚖️

Transforming Individual Points

If you know a point $(x, y)$ on the original graph $y = f(x)$, you can find its image on the transformed graph $y = af(b(x - h)) + k$ using this process:

  1. Horizontal translation: $x_1 = x + h$
  2. Horizontal dilation: $x_2 = \frac{x_1}{b} = \frac{x + h}{b}$
  3. Vertical dilation: $y_2 = a \cdot y$
  4. Vertical translation: $y_3 = y_2 + k = ay + k$

So the final image point is:

$$\left(\frac{x + h}{b}, ay + k\right)$$

Wait — actually, the more natural way to think about it is: if $(x, y)$ is on the original, then for the transformed graph, the point that has the same "relative position" is found by reversing the transformations. Let's be more careful.

If $(x, y)$ is on $y = f(x)$, then to find the corresponding point on $y = af(b(x - h)) + k$:

So the correct transformed point is $\left(\frac{x}{b} + h, ay + k\right)$.

Memory shortcut for points: Divide $x$ by $b$, then add $h$. Multiply $y$ by $a$, then add $k$. This is the opposite order of how you might read the equation, but it is the correct way to map original points to transformed points.

🧮 Worked Examples

Worked Example 1 — Describing Combined Transformations

Stepwise
Describe the transformations that map $y = f(x)$ to $y = -2f(x - 3) + 1$.
  1. 1
    Horizontal translation
    $x - 3$ means shift 3 units to the right.
  2. 2
    Vertical dilation and reflection
    $-2$ outside means vertical dilation by factor 2 and reflection in the $x$-axis.
  3. 3
    Vertical translation
    $+1$ means shift 1 unit up.
✓ Answer Translation 3 units right, vertical dilation by factor 2 from the $x$-axis, reflection in the $x$-axis, translation 1 unit up.

Worked Example 2 — Finding a Transformed Point

Stepwise
The point $(2, 4)$ lies on $y = f(x)$. Find the corresponding point on $y = 3f(2x - 4) + 5$.
  1. 1
    Rewrite in standard form
    y = 3f(2(x - 2)) + 5
    So $a = 3$, $b = 2$, $h = 2$, $k = 5$.
  2. 2
    Transform the $x$-coordinate
    x_{\text{new}} = \frac{x}{b} + h = \frac{2}{2} + 2 = 1 + 2 = 3
  3. 3
    Transform the $y$-coordinate
    y_{\text{new}} = ay + k = 3(4) + 5 = 12 + 5 = 17
✓ Answer $(3, 17)$

Worked Example 3 — Writing the Equation

Stepwise
Write the equation of $y = f(x)$ after the following transformations: reflection in the $y$-axis, horizontal dilation by factor 3, vertical dilation by factor 2, translation 1 unit right and 4 units down.
  1. 1
    Reflection in $y$-axis
    f(-x)
  2. 2
    Horizontal dilation by factor 3
    f\left(-\frac{x}{3}\right)
  3. 3
    Vertical dilation by factor 2
    2f\left(-\frac{x}{3}\right)
  4. 4
    Translation 1 right, 4 down
    y = 2f\left(-\frac{1}{3}(x - 1)\right) - 4
✓ Answer $y = 2f\left(-\dfrac{1}{3}(x - 1)\right) - 4$
⚠️

Common Mistakes — Don't Lose Easy Marks

Reading $f(2x - 4)$ as "dilation by 2 then left 4"
The expression $2x - 4$ is not a dilation by 2 and a translation left 4. It is a dilation by 2 and a translation right 2, because $2x - 4 = 2(x - 2)$. You must factor out the dilation coefficient before reading the translation.
✓ Fix: Always factorise the inside: $f(bx + c) = f(b(x + \frac{c}{b}))$. The translation is $\frac{c}{b}$, not $c$.
Changing the order of transformations incorrectly
When applying transformations to points, some students add the horizontal translation before dividing by $b$, which gives the wrong answer. The correct order is: divide $x$ by $b$ first, then add $h$.
✓ Fix: For points, use the formula $(\frac{x}{b} + h, ay + k)$. Do not reverse the division and addition.
Forgetting to include reflections when $a$ or $b$ is negative
A negative sign in $a$ or $b$ is not just "part of the number" — it is a reflection. $y = -3f(x)$ involves both a vertical dilation by 3 and a reflection in the $x$-axis.
✓ Fix: Always mention the reflection separately when $a < 0$ or $b < 0$.
Confusing the direction of horizontal translations inside factored forms
In $f(b(x - h))$, the translation is $h$ units to the right (because it is $x - h$). Some students see the negative sign and think left. Remember: $x - h$ always shifts right, even when it is inside $b(x - h)$.
✓ Fix: Look only at the sign immediately before $h$. $x - h$ = right. $x + h$ = left.

📓 Copy Into Your Books

▼

📖 General Form

  • $y = af(b(x - h)) + k$
  • $a$ = vertical dilation + $x$-axis reflection if $a < 0$
  • $b$ = horizontal dilation + $y$-axis reflection if $b < 0$

🔢 Point Mapping

  • $x_{\text{new}} = \frac{x}{b} + h$
  • $y_{\text{new}} = ay + k$

⚠️ Factoring Rule

  • Always factor out $b$ from $bx + c$
  • $bx + c = b(x + \frac{c}{b})$
  • The translation is $\frac{c}{b}$, not $c$

💡 Order of Reading

  • Inside out: translation → horizontal dilation → vertical dilation → vertical translation

📝 How are you completing this lesson?

🧪 Activities

🔍 Activity 1 — Calculate + Interpret

Describe the Transformations

For each equation, describe all transformations applied to $y = f(x)$. Be specific about directions, axes, and factors.

  1. 1 $y = 2f(x - 3) + 1$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  2. 2 $y = -f(2x + 4)$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  3. 3 $y = 3f\left(-\frac{x}{2}\right) - 5$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  4. 4 $y = \frac{1}{2}f(x + 1) - 2$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook
🎨 Activity 2 — Predict Coordinates

Find the Transformed Points

The point $(2, 3)$ lies on $y = f(x)$. Find the corresponding point on the graph of each transformed function.

  1. 1 $y = 2f(x - 1) + 4$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  2. 2 $y = -f(2x) + 1$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  3. 3 $y = 3f\left(\frac{x}{2} - 1\right) - 2$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  4. 4 $y = -2f(-x + 3) + 1$

    Type your answer:

    Answer in your workbook.

    Answer in your workbook
Revisit Your Thinking

Earlier you were asked: How would you transform $y = x^2$ so it opens downward, is twice as steep, and has its vertex at $(3, -2)$?

The original parabola $y = x^2$ opens upward with vertex $(0, 0)$. To make it open downward, we need a reflection in the $x$-axis: $-x^2$. To make it twice as steep vertically, we multiply by 2: $-2x^2$. Finally, to move the vertex to $(3, -2)$, we replace $x$ with $(x - 3)$ and subtract 2: $y = -2(x - 3)^2 - 2$. This single equation combines three distinct transformations: reflection, vertical dilation, and translation. That is the power of the general transformation form.

Now revisit your initial response. What did you get right? What has changed in your thinking?

Look back at your initial response in your book. Annotate it with what you now understand differently.

Annotate your initial response in your book
Saved
Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

MC

Multiple Choice

5 random questions from a replayable lesson bank — feedback shown immediately

✍️ Short Answer

📝

Extended Questions

ApplyBand 4

8. The point $(1, 2)$ lies on the graph of $y = f(x)$. Find the corresponding point on the graph of $y = -2f(3x - 6) + 4$. Show all working, including rewriting the function in the form $af(b(x - h)) + k$. 3 MARKS

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook
ApplyBand 4

9. (a) Write the equation of $y = f(x)$ after a reflection in the $x$-axis, a horizontal dilation by factor $\frac{1}{2}$, and a translation 3 units left and 2 units up. (b) If $f(x) = x^2$, simplify your equation from part (a) into expanded polynomial form. 4 MARKS

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook
EvaluateBand 5

10. A student claims that $y = f(2x - 4)$ represents a horizontal dilation by factor $\frac{1}{2}$ followed by a translation 4 units to the left. Evaluate this claim. If it is incorrect, explain the error and state the correct transformations. 3 MARKS

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook

✅ Comprehensive Answers

▼

🔍 Activity 1 — Describe the Transformations Model Answers

1. Vertical dilation by factor 2 from the $x$-axis, translation 3 units right, translation 1 unit up.

2. $2x + 4 = 2(x + 2)$: horizontal dilation by factor $\frac{1}{2}$ from the $y$-axis, translation 2 units left, reflection in the $x$-axis.

3. Horizontal dilation by factor 2 from the $y$-axis, reflection in the $y$-axis, vertical dilation by factor 3 from the $x$-axis, translation 5 units down.

4. Vertical dilation by factor $\frac{1}{2}$ from the $x$-axis, translation 1 unit left, translation 2 units down.

🎨 Activity 2 — Predict Coordinates Model Answers

1. $(3, 10)$ — $x$: $2 + 1 = 3$; $y$: $2(3) + 4 = 10$

2. $(1, -2)$ — $x$: $\frac{2}{2} = 1$; $y$: $-(3) + 1 = -2$

3. $(6, 7)$ — Rewrite: $\frac{x}{2} - 1 = \frac{1}{2}(x - 2)$. $x$: $\frac{2}{\frac{1}{2}} + 2 = 4 + 2 = 6$; wait, using formula: $x_{\text{new}} = \frac{x}{b} + h = \frac{2}{\frac{1}{2}} + 2 = 6$. $y$: $3(3) - 2 = 7$.

4. $(1, -5)$ — Rewrite: $-x + 3 = -(x - 3)$. $x$: $\frac{2}{-1} + 3 = 1$. $y$: $-2(3) + 1 = -5$.

❓ Multiple Choice

1. A — Vertical dilation 2, right 3, up 1.

2. B — $-f(x)$ = $x$-axis reflection; $x + 2$ = left 2.

3. A — Right 1: $(3, 4)$; $\times(-2)$: $(3, -8)$; up 3: $(3, -5)$.

4. B — $y$-axis reflection: $f(-x)$; horizontal dilation 2: $f(-\frac{x}{2})$; up 1: $+1$.

5. A — Right 2, up 5, with $x$-axis reflection. Vertex moves to $(2, 5)$.

📝 Short Answer Model Answers

Q8 (3 marks): $3x - 6 = 3(x - 2)$, so $y = -2f(3(x - 2)) + 4$ [1]. $x_{\text{new}} = \frac{1}{3} + 2 = \frac{7}{3}$ [0.5]. $y_{\text{new}} = -2(2) + 4 = 0$ [1]. New point: $(\frac{7}{3}, 0)$ [0.5].

Q9 (4 marks):

(a) y = -(2x)^2 \text{ with left 3 and up 2} = -4(x + 3)^2 + 2 (b) y = -4(x^2 + 6x + 9) + 2 = -4x^2 - 24x - 36 + 2 = -4x^2 - 24x - 34

Award 2 marks for (a) and 2 marks for correct expansion in (b).

Q10 (3 marks): The student's claim is incorrect [1]. The error is not factoring out the 2: $2x - 4 = 2(x - 2)$, so the translation is 2 units to the right, not 4 units to the left [1]. The correct transformations are: horizontal dilation by factor $\frac{1}{2}$ from the $y$-axis, followed by a translation 2 units to the right [1].

🏎️
Speed Race

Race Through Combined Transformations!

Sprint through questions on sketching graphs with combined transformations. Pool: lessons 1–12.

Mark lesson as complete

Tick when you've finished all activities and checked your answers.

Module overview · Maths Advanced subject page · Next lesson · Module quiz