Introduction to Graph Transformations

Graph transformations allow us to take a known parent function and systematically alter its position, orientation, or scale to produce new functions , without starting from scratch each time.

At AHL level, you need to master four core transformation types and, crucially, how to combine them correctly in composite transformations.

The four core transformations are:

Translations , shifting the graph horizontally or vertically
Reflections , flipping the graph over an axis
Vertical stretches/compressions , scaling the graph away from or towards the x-axis
Horizontal stretches/compressions , scaling the graph away from or towards the y-axis

All of these can be expressed using the general transformed form:

$y = p f (q x - a) + b$

where each parameter controls a specific transformation. Understanding what each parameter does , and in what order to apply them , is the central skill of this subtopic.

Exam Tip

Use $y = x^{2}$ (or $y = sin x$ , $y = ln x$ ) as your go-to parent function when practising transformations. Their familiar shapes make it easy to spot what has changed.

Vertical Translations

Vertical Translation: A transformation of the form $y = f (x) + b$ that shifts every point on the graph of $f$ up or down by $∣ b ∣$ units, without changing the shape of the graph.

The rule is straightforward:

If $b > 0$ : graph shifts up by $b$ units
If $b < 0$ : graph shifts down by $∣ b ∣$ units

Every point $(x, y)$ on $f (x)$ maps to $(x, y + b)$ on the new graph.

Example

Example: Starting from $f (x) = x^{2}$ :

$g (x) = x^{2} + 3$ shifts the parabola 3 units up. The vertex moves from $(0, 0)$ to $(0, 3)$ .
$h (x) = x^{2} - 5$ shifts the parabola 5 units down. The vertex moves to $(0, - 5)$ .

The shape of the parabola is identical in all three cases , only the position changes.

Note

A vertical translation changes the y-intercept and any y-coordinates of key points, but does not affect the x-coordinates of any point, nor does it change the overall shape or orientation of the graph.

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Introduction to Graph Transformations

Graph transformations allow us to take a known parent function and systematically alter its position, orientation, or scale to produce new functions , without starting from scratch each time.

At AHL level, you need to master four core transformation types and, crucially, how to combine them correctly in composite transformations.

The four core transformations are:

Translations , shifting the graph horizontally or vertically
Reflections , flipping the graph over an axis
Vertical stretches/compressions , scaling the graph away from or towards the x-axis
Horizontal stretches/compressions , scaling the graph away from or towards the y-axis

All of these can be expressed using the general transformed form:

$y = p f (q x - a) + b$

where each parameter controls a specific transformation. Understanding what each parameter does , and in what order to apply them , is the central skill of this subtopic.

Exam Tip

Use $y = x^{2}$ (or $y = sin x$ , $y = ln x$ ) as your go-to parent function when practising transformations. Their familiar shapes make it easy to spot what has changed.

Vertical Translations

Vertical Translation: A transformation of the form $y = f (x) + b$ that shifts every point on the graph of $f$ up or down by $∣ b ∣$ units, without changing the shape of the graph.

The rule is straightforward:

If $b > 0$ : graph shifts up by $b$ units
If $b < 0$ : graph shifts down by $∣ b ∣$ units

Every point $(x, y)$ on $f (x)$ maps to $(x, y + b)$ on the new graph.

Example

Example: Starting from $f (x) = x^{2}$ :

$g (x) = x^{2} + 3$ shifts the parabola 3 units up. The vertex moves from $(0, 0)$ to $(0, 3)$ .
$h (x) = x^{2} - 5$ shifts the parabola 5 units down. The vertex moves to $(0, - 5)$ .

The shape of the parabola is identical in all three cases , only the position changes.

Note

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AHL 2.8—Transformations of graphs, composite transformations

Introduction to Graph Transformations

Vertical Translations

12 more sections in this topic

AHL 2.8—Transformations of graphs, composite transformations

Introduction to Graph Transformations

Vertical Translations

12 more sections in this topic