Transformations of functions

IB Mathematics AA HL· 5 min read

1. Vertical and Horizontal Translations★★☆☆☆⏱ 10 min

📘 Definition

Translation

Vertical: $y = f (x) + a$ , Horizontal: $y = f (x + a)$

A translation shifts every point on the graph vertically or horizontally by a constant distance, with no change to shape or orientation.

📐 Worked Example

Given $f (x) = x^{2}$ , write the equation of $f (x)$ translated 3 units up and 2 units to the left.

1
Identify the effect of the vertical translation: 3 units up adds 3 to the entire function, giving $y = f (x) + 3$ .
2
Identify the effect of the horizontal translation: 2 units left means we replace $x$ with $x + 2$ , giving $y = f (x + 2) + 3$ .
3
Substitute $f (x) = x^{2}$ to get the final equation:
4
$y = (x+2)^2 + 3$

Exam tip:

Direction of horizontal translations is the most common exam mistake, always double-check this.

2. Reflections Across Axes★★☆☆☆⏱ 10 min

📘 Definition

Reflection

Over x-axis: $y = - f (x)$ , Over y-axis: $y = f (- x)$

A reflection flips the graph across a line (usually the x-axis or y-axis) to create a mirror image, preserving shape.

📐 Worked Example

Find the coordinates of the y-intercept of $y = - e^{x} + 2$ , starting from $y = e^{x}$ which has y-intercept $(0, 1)$ .

1
Reflection over the x-axis ( $y = - e^{x}$ ) flips the sign of all y-coordinates, so the original intercept $(0, 1)$ becomes $(0, - 1)$ .
2
A vertical translation 2 units up adds 2 to the y-coordinate, so:
3
$(0, -1 + 2) = (0, 1)$

✓ Quick check

What is the equation of $f (x) = sin x$ reflected over the y-axis?

Choose the correct equation
- $y = - sin x$
- $y = sin (- x)$
- $y = sin (x - 1)$
Reveal answer
1 —
Correct! Reflection over the y-axis replaces $x$ with $- x$ .

3. Stretches and Compressions★★★☆☆⏱ 15 min

📘 Definition

Stretch and Compression

Vertical stretch by $a$ : $y = a f (x)$ , Horizontal stretch by $a$ : $y = f (\frac{x}{a})$

A transformation that resizes a graph by a constant scale factor. Scale factor >1 stretches, 0<scale factor<1 compresses.

📐 Worked Example

What is the horizontal scale factor for $g (x) = (2 x)^{3}$ from parent $f (x) = x^{3}$ ?

1
Rewrite $g (x)$ to match the standard form for horizontal stretches: $g (x) = f (2 x) = f (\frac{x}{a})$ .
2
Equate the coefficients of $x$ : $\frac{1}{a} = 2$ , so solve for $a$ :
3
$a = \frac{1}{2}$

Exam tip:

Always confirm if the question asks for vertical or horizontal scale factor, they are often mixed up.

4. Combined Transformations★★★★☆⏱ 20 min

When applying multiple transformations, order matters. The IB exam standard order follows BIDMAS: do stretches/reflections first, then translations, for transformations inside the function (horizontal) and outside (vertical).

📐 Worked Example

Find the equation of $f (x) = x^{2}$ after these transformations in order: 1. Stretch vertically by scale factor 2, 2. Translate 1 unit down, 3. Translate 3 units right.

1
Apply vertical stretch first: multiply $f (x)$ by 2, giving $y = 2 x^{2}$ .
2
Next translate 1 unit down: subtract 1 from the function, giving $y = 2 x^{2} - 1$ .
3
Finally translate 3 units right: replace $x$ with $x - 3$ , giving the final equation:
4
$y = 2(x-3)^2 - 1$

5. Common Pitfalls

Wrong move:

Claiming $f (x + 2)$ shifts the graph 2 units right

Why:

Confuses direction of horizontal transformations, which act directly on the input $x$

Correct move:

$f (x + a)$ shifts left by $a$ units for $a > 0$ , and right for $a < 0$

Wrong move:

Doing translations before stretches when combining transformations

Why:

Reverses the order of operations, leading to an incorrect final equation

Correct move:

Always do stretches/reflections first, then translations for both horizontal and vertical transformations

Wrong move:

Claiming $f (2 x)$ is a horizontal stretch by scale factor 2

Why:

Ignores the inverse relationship for horizontal transformation coefficients

Correct move:

$f (k x)$ for $k > 1$ is a horizontal compression by scale factor $\frac{1}{k}$

Wrong move:

Stretching only positive y-values for a vertical stretch

Why:

Assumes stretching does not affect points below the x-axis

Correct move:

Every y-coordinate is multiplied by the scale factor, so all points are stretched regardless of position

Wrong move:

Reversing the order of inside and outside transformations

Why:

Applies vertical transformations before all horizontal transformations are complete

Correct move:

All transformations inside the function (on $x$ ) are applied first, then transformations outside the function (on $y$ )

6. Quick Reference Cheatsheet

Transformation	Equation	Effect on Graph
Vertical translation	$y = f (x) + a, a > 0$	Shift up $a$ units
Vertical translation	$y = f (x) + a, a < 0$	Shift down $∣ a ∣$ units
Horizontal translation	$y = f (x + a), a > 0$	Shift left $a$ units
Horizontal translation	$y = f (x + a), a < 0$	Shift right $∣ a ∣$ units
Reflection x-axis	$y = - f (x)$	Flip over x-axis, reverse sign of y
Reflection y-axis	$y = f (- x)$	Flip over y-axis, reverse sign of x
Vertical stretch	$y = a f (x), a > 1$	Stretch vertically by $a$
Vertical compression	$y = a f (x), 0 < a < 1$	Compress vertically by $a$
Horizontal stretch	$y = f (\frac{x}{a}), a > 1$	Stretch horizontally by $a$
Horizontal compression	$y = f (a x), a > 1$	Compress horizontally by $\frac{1}{a}$

7. Frequently Asked

Why do horizontal transformations do the opposite of what I expect?

Horizontal transformations act directly on the input $x$ . To get the same output value as the original function, you have to reverse the operation. For example, to get the output $f (5)$ for $f (x + 2)$ , you need to input $x = 3$ , so the point shifts 2 units left.

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.

2025 · 1
Find transformed function equation
2023 · 2
Sketch combined transformation graph
2021 · 1
Order of multiple transformations

Going deeper

syllabusIB AA HL Topic 2 Official SyllabusFull official learning outcomes reference

What's Next

Mastering transformations of functions is a foundational skill used throughout the entire IB AA HL course. You will apply transformations to build complex functions from basic parent functions, and they are critical for graphing polynomials, trigonometric functions, exponentials, logarithms, and reciprocal functions. Transformations frequently appear as part of larger questions on both Paper 1 and Paper 2, so mastering the order of operations and direction of horizontal changes is essential for exam success.