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Module 1 · L12 of 15 ~45 min ⚡ +50 XP in Learn · +25 to complete

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.

Today's hook, 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 $(3, -2)$? Can you write one equation that achieves all three changes?
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Orient and recall

Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
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Recall, your gut answer first
+5 XP warm-up

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.

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Formula reference · this lesson
core notation

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

$a$ = vertical dilation (and $x$-axis reflection if $a < 0$)
$b$ = horizontal dilation (and $y$-axis 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.

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What you'll master
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
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Key terms
Combined transformation
Two or more changes applied to the same graph, one after another. Like this: $y = 2f(x) + 1$ stretches first, then lifts by 1.
Order matters
Doing the same two changes in the other order can give a different graph. Like this: stretching then adding 1 lifts by 1, but adding 1 then stretching lifts by 2.
Factored form
Rewriting the inside of the function so the horizontal change can be read off. Like this: $f(2x - 4)$ becomes $f(2(x - 2))$, so the shift is right 2, not right 4.
Vertical shift
Adding a number outside the function, which moves it up or down. Like this: $y = f(x) - 3$ drops every point by 3.
Horizontal shift
Adding a number inside the function, which moves it the opposite way to the sign. Like this: $y = f(x - 5)$ moves right 5.
Mapping a point
Following one known point through every step of the transformation. Like this: under $y = f(x - 2) + 1$ the point $(0, 0)$ lands at $(2, 1)$.
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Establish it yourself with a graphing application

Work through the core explanation before applying it.

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Establish it yourself with a graphing application
+10 XP

The syllabus dot-point for this lesson begins "Establish using graphing applications", so this is the activity that earns it rather than being told the rule. Use Desmos, GeoGebra or any graphing calculator.

Task. Graph $y = 2(x - 1)^2$, then graph the result of translating $y = x^2$ right by 1 and then dilating vertically by 2, and separately of dilating first and then translating.

Predict before you plot. Before you look: predict whether the two orders give the same graph. Then check. If they differ, work out which order the equation $y = 2(x-1)^2$ actually describes.
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Building the general form
core concept · +3 XP at end

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:

  • $h$: horizontal translation ($h > 0$ means right, $h < 0$ means left)
  • $b$: horizontal dilation by factor $\frac{1}{|b|}$ from the $y$-axis; if $b < 0$, there is also a reflection in the $y$-axis
  • $a$: vertical dilation by factor $|a|$ from the $x$-axis; if $a < 0$, there is also a reflection in the $x$-axis
  • $k$: vertical translation ($k > 0$ means up, $k < 0$ means down)
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:

  • Horizontal and vertical translations commute with each other
  • Horizontal and vertical dilations commute with each other

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.

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

Pause, copy the general transformation form $y = af(b(x-h)) + k$ with the role of each parameter ($a$: vertical dilation/reflection; $b$: horizontal dilation/reflection; $h$: horizontal shift; $k$: vertical shift) into your book.

Did you get this? True or false: in $y = af(b(x - h)) + k$, a negative value of $h$ means the graph shifts to the right.

Quick check: In $y = 3f(2(x - 1)) + 4$, what are the values of $a$, $b$, $h$, and $k$?

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Transforming individual points
core concept

We just saw that $y = af(b(x-h)) + k$ encodes all four transformations in one compact expression. That raises a question: if I know a point $(x_0, y_0)$ on the original graph, where does it land on the transformed graph? This card answers it → the mapping formula $x_{\text{new}} = \frac{x_0}{b} + h$, $y_{\text{new}} = ay_0 + k$.

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 the following formula.

If $(x, y)$ is on $y = f(x)$, the corresponding point on the transformed graph is found by:

  • The input to the transformed function that produces the same inner value is $x_{\text{new}}$ where $b(x_{\text{new}} - h) = x$, so $x_{\text{new}} = \frac{x}{b} + h$
  • The output is $y_{\text{new}} = ay + k$

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.

To map a point $(x, y)$ on $y = f(x)$ to the transformed graph $y = af(b(x-h))+k$:; $x_{\text{new}} = \dfrac{x}{b} + h$ (divide by $b$ first, then add $h$)

Pause, copy the point-mapping formulas: $x_{\text{new}} = \dfrac{x}{b} + h$ and $y_{\text{new}} = ay + k$, and the memory cue (divide $x$ by $b$ first, then add $h$) into your book.

Fill the blanks: drag each token into place.

divide add multiply subtract

To transform an $x$-coordinate: ___ by $b$, then ___ $h$. To transform a $y$-coordinate: ___ by $a$, then add $k$.

Worked examples · reveal as you go

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Work examples end to end

Follow the reasoning through complete worked solutions.

Worked example 1 · describing combined transformations +5 XP on full reveal

Describe the transformations that map $y = f(x)$ to $y = -2f(x - 3) + 1$.

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Identify parameters: $a = -2$, $b = 1$, $h = 3$, $k = 1$
Read from $y = af(b(x - h)) + k$
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Horizontal translation
$x - 3$ means shift 3 units to the right.
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Vertical dilation and reflection
$-2$ outside means vertical dilation by factor 2 and reflection in the $x$-axis.
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Vertical translation
$+1$ means shift 1 unit up.
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Answer: Translation 3 units right, vertical dilation by factor 2, reflection in the $x$-axis, translation 1 unit up. ✓
State all transformations clearly in order.
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Worked example 2 · finding a transformed point +5 XP on full reveal

The point $(2, 4)$ lies on $y = f(x)$. Find the corresponding point on $y = 3f(2x - 4) + 5$.

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Rewrite in standard form: $y = 3f(2(x - 2)) + 5$
So $a = 3$, $b = 2$, $h = 2$, $k = 5$.
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$x_{\text{new}} = \dfrac{x}{b} + h = \dfrac{2}{2} + 2 = 1 + 2 = 3$
Transform the $x$-coordinate: divide by $b$ first, then add $h$.
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$y_{\text{new}} = ay + k = 3(4) + 5 = 12 + 5 = 17$
Transform the $y$-coordinate: multiply by $a$, then add $k$.
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Answer: $(3, 17)$ ✓
State the final transformed point clearly.
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Worked example 3 · writing the equation +5 XP on full reveal

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.

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Reflection in $y$-axis: $f(-x)$
Negate the $x$ inside the function.
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Horizontal dilation by factor 3: $f\!\left(-\dfrac{x}{3}\right)$
Horizontal dilation by factor 3 replaces $x$ with $\frac{x}{3}$.
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Vertical dilation by factor 2: $2f\!\left(-\dfrac{x}{3}\right)$
Multiply the whole function by 2.
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Translation 1 right, 4 down: $y = 2f\!\left(-\dfrac{1}{3}(x - 1)\right) - 4$
Replace $x$ with $(x - 1)$ and subtract 4.
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Answer: $y = 2f\!\left(-\dfrac{1}{3}(x - 1)\right) - 4$ ✓
Final answer in standard form.
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Common mistakes · the 4 traps that cost marks

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Dodge the traps, then apply

Meet the mistakes that cost marks, then do it yourself.

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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 $c$.

2

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.

3

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 $b < 0$.

4

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.

Predict then reveal+8 XP
1 · Predict
2 · Reveal
3 · Compare

Trap 1 says $f(2x - 4)$ is not "left 4". So predict: will $f(2x - 4)$ and $f(2(x - 2))$ draw the same graph or two different graphs, and what translation does $f(2x - 4)$ actually apply?

50%
Interactive · Combined transformations

Try this: press the two comparison buttons, $f(2x - 4)$ then $f(2(x - 2))$, and watch the curves coincide. Then drive the four sliders and read where the anchor point lands, and press Play transformation to see the order the four steps are applied in.

Use the explorer. Load $f(2x - 4)$ and then $f(2(x - 2))$ with the two comparison buttons. True or false: they draw two different graphs.

Use the explorer. Choose the base function $\sqrt{x}$, click the anchor (4, 2) button, then set the sliders to $a = 2$, $b = 2$, $h = 1$, $k = 1$. The panel works the mapping as $4 \div 2 + 1$ and $2 \times 2 + 1$. Read off where the anchor lands: the $x$-coordinate is and the $y$-coordinate is . Then press Challenge me and match a hidden target the same way, without pressing reveal.

Activity 1, Describe the transformations

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

1

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

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

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

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$y = \dfrac{1}{2}f(x + 1) - 2$

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Odd one out: Which equation involves a reflection in the $x$-axis?

Quick-fire practice · 5 reps +2 XP per reveal

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Drill it, then lock it in

Run the quick drill and copy the summary into your book.

1

In $y = af(b(x - h)) + k$, what does a negative value of $a$ indicate?

2

Rewrite $f(3x - 6)$ in standard form $f(b(x - h))$. What are $b$ and $h$?

3

The point $(4, 6)$ is on $y = f(x)$. Find its image on $y = 2f(x) - 3$.

4

Describe the transformations in $y = -f(-x) + 2$.

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Write the equation of $y = f(x)$ after: horizontal dilation by factor $\frac{1}{2}$, vertical dilation by factor 3, translation 2 right and 5 down.

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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.

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Multiple choice

Answer the drill bank and rate your confidence.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

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Short answer

Write full responses, then check them against the model answers.

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Short answer
ApplyBand 43 marks

Q8. 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)

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ApplyBand 44 marks

Q9. (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)

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EvaluateBand 53 marks

Q10. 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)

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📖 Comprehensive answers (click to reveal)

Multiple choice, drill bank

MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.

Drill bank answers:

Reading a combined transformation. Vertical dilation 2, right 3, up 1.

Reflection with a horizontal shift. $-f(x)$ = $x$-axis reflection; $x + 2$ = left 2.

Applying transformations in order. Right 1: $(3, 4)$; $\times(-2)$: $(3, -8)$; up 3: $(3, -5)$.

Building the equation from a description. $y$-axis reflection: $f(-x)$; horizontal dilation 2: $f(-\frac{x}{2})$; up 1: $+1$.

Locating the new vertex. Right 2, up 5, with $x$-axis reflection. Vertex moves to $(2, 5)$.

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.

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: $\left(\frac{7}{3}, 0\right)$ [0.5].

Q9 (4 marks):

(a) $y = -(2x)^2$ 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].

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