Function Transformations: Translations, Reflections & Stretches

What Are Function Transformations?

Function transformations are systematic rules that modify the graph of a function $y = f(x)$ by shifting, reflecting, stretching or compressing it. Instead of plotting dozens of points for each new function, you can start from a known parent graph and apply transformations to obtain the new shape instantly.

This is one of the most powerful techniques in A-Level and AP mathematics — it turns graph sketching from tedious calculation into elegant pattern recognition.

Function Transformation Lab

Drag sliders to apply translations, reflections and stretches to any function in real time. See how each parameter changes the equation and the graph simultaneously.

Launch Transformer

Learning Goals: By the end of this guide, you should be able to:

Apply vertical and horizontal translations to any function graph.

Perform reflections in the x-axis and y-axis.

Apply vertical and horizontal stretches and compressions.

Combine multiple transformations in the correct order.

The Four Families of Transformations

1. Translations (Shifting)

A translation slides the entire graph without changing its shape.

Transformation	Equation	Effect
Vertical shift up by $a$	$y = f(x) + a$	Every point moves up $a$ units
Vertical shift down by $a$	$y = f(x) - a$	Every point moves down $a$ units
Horizontal shift left by $a$	$y = f(x + a)$	Every point moves left $a$ units
Horizontal shift right by $a$	$y = f(x - a)$	Every point moves right $a$ units

The counter-intuitive rule: $f(x + a)$ shifts the graph left, not right. Think of it as: the function reaches each value $a$ units earlier along the x-axis.

2. Reflections

Transformation	Equation	Effect
Reflect in x-axis	$y = -f(x)$	All y-coordinates are negated
Reflect in y-axis	$y = f(-x)$	All x-coordinates are negated

3. Stretches and Compressions

Transformation	Equation	Effect
Vertical stretch by factor $a$	$y = af(x)$	y-coordinates multiplied by $a$
Vertical compression by factor $a$	$y = \frac{1}{a}f(x)$	y-coordinates divided by $a$
Horizontal compression by factor $a$	$y = f(ax)$	x-coordinates divided by $a$
Horizontal stretch by factor $a$	$y = f\left(\frac{x}{a}\right)$	x-coordinates multiplied by $a$

Another counter-intuitive rule: $f(2x)$ makes the graph narrower (compressed horizontally), not wider. It's the opposite of what the "2" might suggest.

4. Combined Transformations

When multiple transformations are applied, the order matters. The general form is:

y = a \cdot f(b(x - h)) + k

Where:

$a$ = vertical stretch/reflection
$b$ = horizontal stretch/reflection
$h$ = horizontal translation
$k$ = vertical translation

Recommended order: Horizontal operations first (inside the function), then vertical operations (outside).

Worked Examples

Example 1: Describing a Transformation

Question: The graph of $y = x^2$ is transformed to $y = (x - 3)^2 + 2$ . Describe the transformation.

Solution: Comparing with $y = f(x - h) + k$ , we identify $h = 3$ and $k = 2$ .

The graph is translated 3 units to the right and 2 units up. The vertex moves from $(0, 0)$ to $(3, 2)$ .

Example 2: Finding the New Equation

Question: The graph of $y = \sin x$ is reflected in the x-axis and then stretched vertically by a factor of 3. Write the equation of the new graph.

Step 1: Reflect in x-axis: $y = -\sin x$

Step 2: Vertical stretch by factor 3: $y = 3(-\sin x) = -3\sin x$

Answer: $y = -3\sin x$

Common Mistakes

Confusing the direction of horizontal transformations — $f(x + 2)$ shifts LEFT, not right. $f(x - 2)$ shifts RIGHT, not left.
Wrong order of combined transformations — Always apply transformations inside the brackets first (horizontal), then outside (vertical).
Forgetting that horizontal stretches are reciprocal — $f(2x)$ is a horizontal compression by factor $\frac{1}{2}$ , not a stretch by factor 2.

Exam Tips (A-Level / AP / IB)

When asked to "describe the transformation", always state the type (translation/reflection/stretch) and the details (direction, distance, scale factor, axis of reflection).
For translations, use vector notation where expected: a translation of $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ means 3 right, 2 down.
Check your answer by substituting a known point from the original graph and verifying it maps to the correct point on the transformed graph.

Frequently Asked Questions

Does the order of transformations always matter?

Not always — commutative pairs like two translations or two reflections in different axes can be swapped. But mixing translations with stretches or reflections does depend on order. When in doubt, follow: horizontal operations first, vertical second.

How do I handle transformations of trigonometric functions?

The same rules apply. For $y = A\sin(B(x - C)) + D$ : $A$ is the amplitude (vertical stretch), $B$ affects the period ( $\text{period} = \frac{2\pi}{B}$ ), $C$ is the phase shift, and $D$ is the vertical translation.

Trigonometric Graphs — See how transformations affect the shape and position of sine and cosine curves.
Derivatives from First Principles — Understand how transformations affect the derivative of a function.
Conic Sections — Apply translations to reposition circles, ellipses and parabolas.

Try the interactive simulation for this topic