Transformations of function graphs

IB Mathematics AI SL· Unit 2: Functions, Topic 7· 15 min read

1. Vertical and Horizontal Translations★★☆☆☆⏱ 5 min

📘 Definition

Translation

A rigid transformation that shifts every point on the graph of a function by a fixed distance, with no change to shape or size of the graph.

Translations follow consistent rules based on whether the constant is added inside or outside the function argument.

For original function $y = f (x)$ , a vertical translation by $a$ units is $y = f (x) + a$ . Positive $a$ shifts up, negative $a$ shifts down.
A horizontal translation by $b$ units is $y = f (x - b)$ . Positive $b$ shifts right, negative $b$ shifts left.

📐 Worked Example

Original function is $f (x) = x^{2}$ . Write the equation of $f (x)$ translated 2 units down and 4 units right.

1
Start with the original function:
$f(x) = x^2$
2
Apply horizontal translation 4 units right: replace $x$ with $x - 4$ :
$y = f(x - 4) = (x - 4)^2$
3
Apply vertical translation 2 units down: add -2 outside the function:
$y = (x - 4)^2 - 2$

Exam tip:

Always confirm your translation by checking a key point (like a vertex) to confirm shift direction.

2. Reflections Across the Axes★★☆☆☆⏱ 4 min

📘 Definition

Reflection

A transformation that flips the graph across a given line to create a mirror image. Shape is unchanged, but orientation is reversed.

Reflections across the x-axis and y-axis follow simple rules based on where the negative sign is placed.

Reflection across the x-axis: $y = - f (x)$ : every y-coordinate is multiplied by -1.
Reflection across the y-axis: $y = f (- x)$ : every x-coordinate is multiplied by -1.

📐 Worked Example

Original $f (x) = e^{x}$ has y-intercept at $(0, 1)$ . Find the y-intercept of $y = - f (x)$ reflected across the x-axis.

1
Apply the reflection rule: all y-coordinates are multiplied by -1.
2
Original y-intercept is $(0, 1)$ , so multiply the y-coordinate by -1:
$(0, 1 \times -1) = (0, -1)$

3. Vertical and Horizontal Dilations★★★☆☆⏱ 6 min

📘 Definition

Dilation

A non-rigid transformation that stretches or compresses a graph by a scale factor, changing size but not overall shape.

The location of the scale factor (inside or outside the function) tells you if the dilation is horizontal or vertical, just like translations.

Vertical dilation by scale factor $k$ : $y = k f (x)$ , all y-coordinates multiplied by $k$ .
Horizontal dilation by scale factor $\frac{1}{c}$ : $y = f (c x)$ , all x-coordinates multiplied by $\frac{1}{c}$ .

📐 Worked Example

Given $f (x) = sin x$ , find the equation after a vertical stretch by scale factor 3 and horizontal compression by scale factor 2.

1
Apply vertical stretch first: multiply the entire function by 3:
$y = 3 f(x) = 3 \sin x$
2
Apply horizontal compression by 2: replace $x$ with $2 x$ :
$y = 3 \sin(2x)$
3
Verify: original period of $sin x$ is $2 π$ , new period is $\frac{2 π}{2} = π$ , which matches the compression.

Exam tip:

Dilations from the x-axis are vertical, dilations from the y-axis are horizontal.

4. Combined Transformations★★★★☆⏱ 6 min

When multiple transformations are applied, order matters. We follow an order consistent with standard order of operations (BIDMAS/BODMAS):

First complete all horizontal transformations (inside $f (...)$ ): do dilations/reflections first, then translations.
Then complete all vertical transformations (outside $f (...)$ ): do dilations/reflections first, then translations.

📐 Worked Example

Starting with $f (x) = x^{2}$ , write the equation after: reflection across x-axis, horizontal shift 1 unit left, vertical shift 3 units up.

1
Start with original function:
$y = x^2$
2
Apply horizontal transformation: shift 1 unit left: replace $x$ with $x + 1$ :
$y = (x + 1)^2$
3
Apply vertical transformation: reflection across x-axis, multiply by -1:
$y = -(x + 1)^2$
4
Apply vertical translation 3 units up: add 3:
$y = -(x + 1)^2 + 3$
5
Check: original vertex at $(0, 0)$ , new vertex at $(- 1, 3)$ , which matches the required transformations.

✓ Quick check

Test your understanding of order:

What is the correct order to get $y = 2 f (x + 3) - 1$ from $y = f (x)$ ?
- A: Shift left 3, vertical stretch by 2, shift down 1
- B: Vertical stretch by 2, shift left 3, shift down 1
- C: Shift down 1, vertical stretch by 2, shift left 3
Reveal answer
A —
Correct! Horizontal transformations are done first, then vertical transformations with dilations before translations.

5. Common Pitfalls

Wrong move:

Writing $f (x + 2)$ as a shift 2 units right

Why:

The rule for horizontal translation is $f (x - b)$ for shift $b$ , so the sign is reversed

Correct move:

Recognize $f (x + c)$ shifts $c$ units left, $f (x - c)$ shifts $c$ units right for positive $c$

Wrong move:

Stretching $f (2 x)$ horizontally by factor 2

Why:

The scale factor for horizontal dilations is inverted

Correct move:

$f (c x)$ dilates horizontally by factor $\frac{1}{c}$ , so $f (2 x)$ compresses horizontally by factor 2

Wrong move:

Doing vertical translation before vertical dilation

Why:

Order of operations requires multiplication (dilation) before addition (translation)

Correct move:

Always apply dilations and reflections before translations for vertical transformations

Wrong move:

Interpreting $f (- x)$ as a reflection across the x-axis

Why:

The negative sign is inside the function argument, so it affects x-coordinates not y-coordinates

Correct move:

Negative outside $(- f (x))$ = x-axis reflection; negative inside $(f (- x))$ = y-axis reflection

Wrong move:

Interpreting $y = f (x) + a$ as a horizontal shift left by $a$

Why:

Confusing the position of the constant inside vs outside the function

Correct move:

Constants outside the function change y-coordinates, so result in a vertical shift

6. Quick Reference Cheatsheet

Transformation	Rule for $y = f (x)$	Effect on Graph
Vertical translation $a$	$y = f (x) + a$	Shift up $a$ units ( $a > 0$ )
Horizontal translation $b$	$y = f (x - b)$	Shift right $b$ units ( $b > 0$ )
Reflection across x-axis	$y = - f (x)$	Flip over x-axis, $y \to - y$
Reflection across y-axis	$y = f (- x)$	Flip over y-axis, $x \to - x$
Vertical dilation $k$	$y = k f (x)$	Stretch vertically by factor $k$
Horizontal dilation $c$	$y = f (c x)$	Compress horizontally by factor $c$

7. Frequently Asked

Do I need to sketch transformed graphs in exams?

Yes, both Paper 1 (non-calculator) and Paper 2 (calculator) regularly require accurate sketches of transformed functions, including labeling key points.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Find transformed function equation
2021 · 2
Sketch transformed exponential graph
2023 · 1
Identify transformation from equation

Going deeper

syllabusIB Official Math AI SL SyllabusReview Unit 2 learning outcomes

What's Next

Mastering function graph transformations is a foundational skill for nearly all graphing topics in IB Math AI SL, including quadratic functions, exponential and logarithmic models, and periodic trigonometric functions. Transformations allow you to quickly sketch complex graphs starting from simple parent functions, which saves critical time in both Paper 1 (non-calculator) and Paper 2 (calculator) exams. You will also apply transformations to model real-world phenomena, from shifting seasonal temperature models to scaling population growth curves. This knowledge directly prepares you for working with more complex functions and modeling problems that make up a large portion of your final exam.