Transforming Graphs y=f(x)

What you'll learn

How to understand the original graph written as $y = f (x)$ .
How changes outside the bracket affect the $y$ -values.
How changes inside the bracket affect the $x$ -values — often the opposite of what you expect.
How to describe translations, stretches and reflections clearly in exam language.

The starting point: $y = f (x)$

Before transforming graphs, you need to be comfortable with function notation.

Definition

Function notation

$f (x)$ means “the value of the function when the input is $x$ ”. The graph $y = f (x)$ is the original graph before any transformation has happened.

Think of the graph as a collection of points. If the original graph $y = f (x)$ contains the point $(2, 5)$ , that means when $x = 2$ , the output is $y = 5$ .

A transformation changes where the points go. Instead of redrawing everything from scratch, you can track how each coordinate changes.

Key Idea

Point tracking

A graph transformation can be understood by asking: “What happens to every point $(x, y)$ on the original graph?”

Example

Reading a point from $y = f (x)$

The graph $y = f (x)$ passes through the points $(- 2, 4)$ , $(0, 1)$ and $(3, 6)$ . What does this tell us about $f (3)$ ?

Find the point where the $x$ -coordinate is 3.
The matching $y$ -coordinate is 6.
Therefore, $f (3) = 6$ .

The golden rule

There are two main places a transformation can appear:

Outside the function, such as $y = f (x) + 3$ or $y = 2 f (x)$ .
Inside the bracket, such as $y = f (x - 4)$ or $y = f (3 x)$ .

Key Idea

Outside vs inside

Changes outside the bracket affect the $y$ -values and do what they say. Changes inside the bracket affect the $x$ -values and usually do the opposite.

These transformations are easiest to understand by comparing the original graph with the transformed graph.

Original parabola compared with a right shift and an upward shift

Vertical translations: $y = f (x) + a$

A translation is a slide of a graph. Its shape and size do not change.

Definition

Vertical translation

A vertical translation moves every point up or down. In $y = f (x) + a$ , add $a$ to every $y$ -value.

For example:

$y = f (x) + 4$ moves the graph up 4.
$y = f (x) - 7$ moves the graph down 7.

The $x$ -coordinates stay the same.

Example

Moving a graph upwards

A point on $y = f (x)$ is $(2, - 3)$ . Find where this point moves on $y = f (x) + 5$ .

The transformation is outside the bracket, so it changes the $y$ -value.
It says +5, so add 5 to the original $y$ -coordinate.
The $x$ -coordinate stays as 2.
The new $y$ -coordinate is $- 3 + 5 = 2$ .
The point moves to $(2, 2)$ .

Common Mistake

Changing the wrong coordinate

For $y = f (x) + 5$ , do not add 5 to the $x$ -coordinate. The +5 is outside the bracket, so it changes the $y$ -values only.

Vertical stretches and reflections: $y = a f (x)$

A stretch makes a graph wider, narrower, taller or shorter. For $y = a f (x)$ , the multiplier is outside the function, so it changes the $y$ -values.

Definition

Vertical stretch

The graph $y = a f (x)$ multiplies every $y$ -value by $a$ . This is a vertical stretch with scale factor $a$ .

For example:

$y = 3 f (x)$ multiplies all $y$ -values by 3.
$y=12f(x)y=\frac{1}{2}f(x)$ halves all $y$ -values.
$y = - f (x)$ multiplies all $y$ -values by -1, giving a reflection in the $x$ -axis.

Definition

Reflection

A reflection flips a graph in a mirror line. The graph $y = - f (x)$ is a reflection of $y = f (x)$ in the $x$ -axis.

Example

Vertical stretch from coordinates

The graph $y = f (x)$ passes through $(- 1, 4)$ , $(0, - 2)$ and $(3, 5)$ . Find the matching points on $y = 2 f (x)$ .

The multiplier 2 is outside the bracket, so it affects the $y$ -values.
Keep each $x$ -coordinate the same.
Double each $y$ -coordinate: 4 becomes 8, -2 becomes -4, and 5 becomes 10.
The transformed points are $(- 1, 8)$ , $(0, - 4)$ and $(3, 10)$ .

Example

Reflection in the x-axis

A point on $y = f (x)$ is $(- 4, 7)$ . Find where it moves on $y = - f (x)$ .

The negative sign is outside the bracket, so it changes the $y$ -value.
Multiplying the $y$ -coordinate by -1 changes 7 to -7.
The $x$ -coordinate stays as -4.
The point moves to $(- 4, - 7)$ .

Original parabola compared with vertical stretch, horizontal compression and reflection

Horizontal translations: $y = f (x - a)$

Now comes the part students often find strange.

When the change is inside the bracket, it affects the $x$ -values. But it does the opposite.

Definition

Horizontal translation

A horizontal translation moves every point left or right. In $y = f (x - a)$ , the graph moves right by $a$ . In $y = f (x + a)$ , the graph moves left by $a$ .

Examples:

$y = f (x - 6)$ moves the graph right 6.
$y = f (x + 2)$ moves the graph left 2.

Why is $x - 6$ a move right? Because to get the same input as before, the new $x$ -value must be 6 bigger.

Example

Moving a graph to the right

A point on $y = f (x)$ is $(1, 5)$ . Find where it moves on $y = f (x - 4)$ .

The change is inside the bracket, so it affects the $x$ -value.
Inside the bracket says -4, so do the opposite: add 4 to the $x$ -coordinate.
The new $x$ -coordinate is $1 + 4 = 5$ .
The $y$ -coordinate stays as 5.
The point moves to $(5, 5)$ .

Example

Moving a graph to the left

A point on $y = f (x)$ is $(- 3, 2)$ . Find where it moves on $y = f (x + 7)$ .

The change is inside the bracket, so it affects the $x$ -value.
Inside the bracket says +7, so do the opposite: subtract 7 from the $x$ -coordinate.
The new $x$ -coordinate is $- 3 - 7 = - 10$ .
The $y$ -coordinate stays as 2.
The point moves to $(- 10, 2)$ .

Tip

Quick memory trick

Outside: $y$ does what it says. Inside: $x$ does the opposite.

Horizontal stretches and compressions: $y = f (k x)$

A change like $y = f (2 x)$ is also inside the bracket, so it affects the $x$ -values.

Definition

Horizontal stretch or compression

The graph $y = f (k x)$ changes every $x$ -value by dividing it by $k$ . So $y = f (2 x)$ halves the $x$ -values, and $y=f(13x)y=f(\frac{1}{3}x)$ triples the $x$ -values.

This is why $y = f (2 x)$ looks narrower: every point is pulled closer to the $y$ -axis.

Example

Horizontal compression

A point on $y = f (x)$ is $(8, 3)$ . Find where it moves on $y = f (2 x)$ .

The 2 is inside the bracket, so it affects the $x$ -value.
Because inside changes do the opposite, divide the $x$ -coordinate by 2.
The new $x$ -coordinate is $\div 2=4$ .
The $y$ -coordinate stays as 3.
The point moves to $(4, 3)$ .

Example

Horizontal stretch

A point on $y = f (x)$ is $(- 2, 6)$ . Find where it moves on $y=f(14x)y=f(\frac{1}{4}x)$ .

The multiplier $14\frac{1}{4}$ is inside the bracket, so it affects the $x$ -value.
Divide by $14\frac{1}{4}$ , which is the same as multiplying by 4.
The new $x$ -coordinate is $\times 4=-8$ .
The $y$ -coordinate stays as 6.
The point moves to $(- 8, 6)$ .

Common Mistake

Multiplying instead of dividing

For $y = f (3 x)$ , the $x$ -values are divided by 3, not multiplied by 3. Inside the bracket means opposite effect.

Reflections in the axes

There are two common reflections you need to recognise.

Reflection in the $x$ -axis: $y = - f (x)$

This is outside the bracket, so it changes the $y$ -values. Every $y$ -coordinate is multiplied by -1.

A point $(x, y)$ becomes $(x, - y)$ .

Reflection in the $y$ -axis: $y = f (- x)$

This is inside the bracket, so it changes the $x$ -values. Every $x$ -coordinate is multiplied by -1.

A point $(x, y)$ becomes $(- x, y)$ .

Example

Choosing the correct mirror line

A point on $y = f (x)$ is $(6, - 4)$ . Find where it moves on $y = f (- x)$ .

The negative sign is inside the bracket, so it affects the $x$ -value.
Multiply the $x$ -coordinate by -1: 6 becomes -6.
The $y$ -coordinate stays as -4.
The point moves to $(- 6, - 4)$ .
This is a reflection in the $y$ -axis.

Combining transformations

At Grade 8/9, you may see more than one transformation at once, such as $y = 3 f (x - 2) - 1$ .

Handle this carefully:

Inside the bracket affects $x$ .
Outside the bracket affects $y$ .
Work with coordinates if the graph has key points.

Example

Combining a horizontal shift and vertical change

A point on $y = f (x)$ is $(4, 5)$ . Find where it moves on $y = 2 f (x - 3) - 7$ .

Look inside the bracket first: $x - 3$ means the $x$ -values increase by 3.
The new $x$ -coordinate is $4 + 3 = 7$ .
Now look outside the bracket: $2 f (x - 3) - 7$ means double the $y$ -value, then subtract 7.
The new $y$ -coordinate is $\times 5-7=3$ .
The point moves to $(7, 3)$ .

Common Mistake

Order matters with vertical changes

In $y = 2 f (x) - 7$ , the original $y$ -values are multiplied by 2 first, then 7 is subtracted. So a point with $y = 5$ becomes 3, not -4.

Describing transformations in words

In exams, you may be asked to describe the transformation fully. Be precise.

Good descriptions include:

the type of transformation: translation, stretch, compression or reflection;
the direction or axis, if needed;
the number of units or scale factor.

Example

Describing transformations

Describe the transformation from $y = f (x)$ to $y = f (x + 5)$ .

The change is inside the bracket, so it affects the $x$ -values.
Inside says +5, so do the opposite: move left 5.
The transformation is a translation 5 units to the left.

Example

Using the correct scale factor

Describe the transformation from $y = f (x)$ to $y = f (4 x)$ .

The 4 is inside the bracket, so it affects the $x$ -values.
The $x$ -values are divided by 4.
The graph is compressed horizontally by scale factor $14\frac{1}{4}$ .

Exam technique

In the exam

First decide whether the change is outside or inside the bracket.
For outside changes, alter the $y$ -values exactly as written.
For inside changes, alter the $x$ -values using the opposite effect.
If stuck, pick one clear point on the graph and track where it moves.

Self review

Check yourself

What happens to the graph of $y = f (x)$ when it becomes $y = f (x - 8)$ ?
A point $(3, - 2)$ lies on $y = f (x)$ . Where does it move on $y = - 3 f (x)$ ?
Why does $y = f (5 x)$ make the graph narrower rather than wider?

Transforming Graphs y=f(x)

What you'll learn

How to understand the original graph written as $y = f (x)$ .
How changes outside the bracket affect the $y$ -values.
How changes inside the bracket affect the $x$ -values — often the opposite of what you expect.
How to describe translations, stretches and reflections clearly in exam language.

The starting point: $y = f (x)$

Before transforming graphs, you need to be comfortable with function notation.

Definition

Function notation

$f (x)$ means “the value of the function when the input is $x$ ”. The graph $y = f (x)$ is the original graph before any transformation has happened.

Think of the graph as a collection of points. If the original graph $y = f (x)$ contains the point $(2, 5)$ , that means when $x = 2$ , the output is $y = 5$ .

A transformation changes where the points go. Instead of redrawing everything from scratch, you can track how each coordinate changes.

Key Idea

Point tracking

A graph transformation can be understood by asking: “What happens to every point $(x, y)$ on the original graph?”

Example

Reading a point from $y = f (x)$

The graph $y = f (x)$ passes through the points $(- 2, 4)$ , $(0, 1)$ and $(3, 6)$ . What does this tell us about $f (3)$ ?

Find the point where the $x$ -coordinate is 3.
The matching $y$ -coordinate is 6.
Therefore, $f (3) = 6$ .

The golden rule

There are two main places a transformation can appear:

Outside the function, such as $y = f (x) + 3$ or $y = 2 f (x)$ .
Inside the bracket, such as $y = f (x - 4)$ or $y = f (3 x)$ .

Key Idea

Outside vs inside

Changes outside the bracket affect the $y$ -values and do what they say. Changes inside the bracket affect the $x$ -values and usually do the opposite.

These transformations are easiest to understand by comparing the original graph with the transformed graph.

Original parabola compared with a right shift and an upward shift

Vertical translations: $y = f (x) + a$

A translation is a slide of a graph. Its shape and size do not change.

Definition

Vertical translation

A vertical translation moves every point up or down. In $y = f (x) + a$ , add $a$ to every $y$ -value.

For example:

$y = f (x) + 4$ moves the graph up 4.
$y = f (x) - 7$ moves the graph down 7.

The $x$ -coordinates stay the same.

Example

Moving a graph upwards

A point on $y = f (x)$ is $(2, - 3)$ . Find where this point moves on $y = f (x) + 5$ .

The transformation is outside the bracket, so it changes the $y$ -value.
It says +5, so add 5 to the original $y$ -coordinate.
The $x$ -coordinate stays as 2.
The new $y$ -coordinate is $- 3 + 5 = 2$ .
The point moves to $(2, 2)$ .

Common Mistake

Changing the wrong coordinate

For $y = f (x) + 5$ , do not add 5 to the $x$ -coordinate. The +5 is outside the bracket, so it changes the $y$ -values only.

Vertical stretches and reflections: $y = a f (x)$

A stretch makes a graph wider, narrower, taller or shorter. For $y = a f (x)$ , the multiplier is outside the function, so it changes the $y$ -values.

Definition

Vertical stretch

The graph $y = a f (x)$ multiplies every $y$ -value by $a$ . This is a vertical stretch with scale factor $a$ .

For example:

$y = 3 f (x)$ multiplies all $y$ -values by 3.
$y=12f(x)y=\frac{1}{2}f(x)$ halves all $y$ -values.
$y = - f (x)$ multiplies all $y$ -values by -1, giving a reflection in the $x$ -axis.

Definition

Reflection

A reflection flips a graph in a mirror line. The graph $y = - f (x)$ is a reflection of $y = f (x)$ in the $x$ -axis.

Example

Vertical stretch from coordinates

The graph $y = f (x)$ passes through $(- 1, 4)$ , $(0, - 2)$ and $(3, 5)$ . Find the matching points on $y = 2 f (x)$ .

The multiplier 2 is outside the bracket, so it affects the $y$ -values.
Keep each $x$ -coordinate the same.
Double each $y$ -coordinate: 4 becomes 8, -2 becomes -4, and 5 becomes 10.
The transformed points are $(- 1, 8)$ , $(0, - 4)$ and $(3, 10)$ .

Example

Reflection in the x-axis

A point on $y = f (x)$ is $(- 4, 7)$ . Find where it moves on $y = - f (x)$ .

The negative sign is outside the bracket, so it changes the $y$ -value.
Multiplying the $y$ -coordinate by -1 changes 7 to -7.
The $x$ -coordinate stays as -4.
The point moves to $(- 4, - 7)$ .

Original parabola compared with vertical stretch, horizontal compression and reflection

Horizontal translations: $y = f (x - a)$

Now comes the part students often find strange.

When the change is inside the bracket, it affects the $x$ -values. But it does the opposite.

Definition

Horizontal translation

A horizontal translation moves every point left or right. In $y = f (x - a)$ , the graph moves right by $a$ . In $y = f (x + a)$ , the graph moves left by $a$ .

Examples:

$y = f (x - 6)$ moves the graph right 6.
$y = f (x + 2)$ moves the graph left 2.

Why is $x - 6$ a move right? Because to get the same input as before, the new $x$ -value must be 6 bigger.

Example

Moving a graph to the right

A point on $y = f (x)$ is $(1, 5)$ . Find where it moves on $y = f (x - 4)$ .

The change is inside the bracket, so it affects the $x$ -value.
Inside the bracket says -4, so do the opposite: add 4 to the $x$ -coordinate.
The new $x$ -coordinate is $1 + 4 = 5$ .
The $y$ -coordinate stays as 5.
The point moves to $(5, 5)$ .

Example

Moving a graph to the left

A point on $y = f (x)$ is $(- 3, 2)$ . Find where it moves on $y = f (x + 7)$ .

The change is inside the bracket, so it affects the $x$ -value.
Inside the bracket says +7, so do the opposite: subtract 7 from the $x$ -coordinate.
The new $x$ -coordinate is $- 3 - 7 = - 10$ .
The $y$ -coordinate stays as 2.
The point moves to $(- 10, 2)$ .

Tip

Quick memory trick

Outside: $y$ does what it says. Inside: $x$ does the opposite.

Horizontal stretches and compressions: $y = f (k x)$

A change like $y = f (2 x)$ is also inside the bracket, so it affects the $x$ -values.

Definition

Horizontal stretch or compression

The graph $y = f (k x)$ changes every $x$ -value by dividing it by $k$ . So $y = f (2 x)$ halves the $x$ -values, and $y=f(13x)y=f(\frac{1}{3}x)$ triples the $x$ -values.

This is why $y = f (2 x)$ looks narrower: every point is pulled closer to the $y$ -axis.

Example

Horizontal compression

A point on $y = f (x)$ is $(8, 3)$ . Find where it moves on $y = f (2 x)$ .

The 2 is inside the bracket, so it affects the $x$ -value.
Because inside changes do the opposite, divide the $x$ -coordinate by 2.
The new $x$ -coordinate is $\div 2=4$ .
The $y$ -coordinate stays as 3.
The point moves to $(4, 3)$ .

Example

Horizontal stretch

A point on $y = f (x)$ is $(- 2, 6)$ . Find where it moves on $y=f(14x)y=f(\frac{1}{4}x)$ .

The multiplier $14\frac{1}{4}$ is inside the bracket, so it affects the $x$ -value.
Divide by $14\frac{1}{4}$ , which is the same as multiplying by 4.
The new $x$ -coordinate is $\times 4=-8$ .
The $y$ -coordinate stays as 6.
The point moves to $(- 8, 6)$ .

Common Mistake

Multiplying instead of dividing

For $y = f (3 x)$ , the $x$ -values are divided by 3, not multiplied by 3. Inside the bracket means opposite effect.

Reflections in the axes

There are two common reflections you need to recognise.

Reflection in the $x$ -axis: $y = - f (x)$

This is outside the bracket, so it changes the $y$ -values. Every $y$ -coordinate is multiplied by -1.

A point $(x, y)$ becomes $(x, - y)$ .

Reflection in the $y$ -axis: $y = f (- x)$

This is inside the bracket, so it changes the $x$ -values. Every $x$ -coordinate is multiplied by -1.

A point $(x, y)$ becomes $(- x, y)$ .

Example

Choosing the correct mirror line

A point on $y = f (x)$ is $(6, - 4)$ . Find where it moves on $y = f (- x)$ .

The negative sign is inside the bracket, so it affects the $x$ -value.
Multiply the $x$ -coordinate by -1: 6 becomes -6.
The $y$ -coordinate stays as -4.
The point moves to $(- 6, - 4)$ .
This is a reflection in the $y$ -axis.

Combining transformations

At Grade 8/9, you may see more than one transformation at once, such as $y = 3 f (x - 2) - 1$ .

Handle this carefully:

Inside the bracket affects $x$ .
Outside the bracket affects $y$ .
Work with coordinates if the graph has key points.

Example

Combining a horizontal shift and vertical change

A point on $y = f (x)$ is $(4, 5)$ . Find where it moves on $y = 2 f (x - 3) - 7$ .

Look inside the bracket first: $x - 3$ means the $x$ -values increase by 3.
The new $x$ -coordinate is $4 + 3 = 7$ .
Now look outside the bracket: $2 f (x - 3) - 7$ means double the $y$ -value, then subtract 7.
The new $y$ -coordinate is $\times 5-7=3$ .
The point moves to $(7, 3)$ .

Common Mistake

Order matters with vertical changes

In $y = 2 f (x) - 7$ , the original $y$ -values are multiplied by 2 first, then 7 is subtracted. So a point with $y = 5$ becomes 3, not -4.

Describing transformations in words

In exams, you may be asked to describe the transformation fully. Be precise.

Good descriptions include:

the type of transformation: translation, stretch, compression or reflection;
the direction or axis, if needed;
the number of units or scale factor.

Example

Describing transformations

Describe the transformation from $y = f (x)$ to $y = f (x + 5)$ .

The change is inside the bracket, so it affects the $x$ -values.
Inside says +5, so do the opposite: move left 5.
The transformation is a translation 5 units to the left.

Example

Using the correct scale factor

Describe the transformation from $y = f (x)$ to $y = f (4 x)$ .

The 4 is inside the bracket, so it affects the $x$ -values.
The $x$ -values are divided by 4.
The graph is compressed horizontally by scale factor $14\frac{1}{4}$ .

Exam technique

In the exam

First decide whether the change is outside or inside the bracket.
For outside changes, alter the $y$ -values exactly as written.
For inside changes, alter the $x$ -values using the opposite effect.
If stuck, pick one clear point on the graph and track where it moves.

Self review

Check yourself

What happens to the graph of $y = f (x)$ when it becomes $y = f (x - 8)$ ?
A point $(3, - 2)$ lies on $y = f (x)$ . Where does it move on $y = - 3 f (x)$ ?
Why does $y = f (5 x)$ make the graph narrower rather than wider?

Transforming Graphs y=f(x)

What you'll learn

The starting point: $y = f (x)$

The golden rule

Vertical translations: $y = f (x) + a$

Vertical stretches and reflections: $y = a f (x)$

Horizontal translations: $y = f (x - a)$

Horizontal stretches and compressions: $y = f (k x)$

Reflections in the axes

Reflection in the $x$ -axis: $y = - f (x)$

Reflection in the $y$ -axis: $y = f (- x)$

Combining transformations

Describing transformations in words

Recap questions

You've reached the end

Transforming Graphs y=f(x)

What you'll learn

The starting point: $y = f (x)$

The golden rule

Vertical translations: $y = f (x) + a$

Vertical stretches and reflections: $y = a f (x)$

Horizontal translations: $y = f (x - a)$

Horizontal stretches and compressions: $y = f (k x)$

Reflections in the axes

Reflection in the $x$ -axis: $y = - f (x)$

Reflection in the $y$ -axis: $y = f (- x)$

Combining transformations

Describing transformations in words

Recap questions

You've reached the end

Transforming Graphs y=f(x)

What you'll learn

The starting point: y=f(x)y=f(x)y=f(x)

The golden rule

Vertical translations: y=f(x)+ay=f(x)+ay=f(x)+a

Vertical stretches and reflections: y=af(x)y=af(x)y=af(x)

Horizontal translations: y=f(x−a)y=f(x-a)y=f(x−a)

Horizontal stretches and compressions: y=f(kx)y=f(kx)y=f(kx)

Reflections in the axes

Reflection in the xxx-axis: y=−f(x)y=-f(x)y=−f(x)

Reflection in the yyy-axis: y=f(−x)y=f(-x)y=f(−x)

Combining transformations

Describing transformations in words

Recap questions

You've reached the end

Transforming Graphs y=f(x) Revision Guide

Transforming Graphs y=f(x)

What you'll learn

The starting point: y=f(x)y=f(x)y=f(x)

The golden rule

Vertical translations: y=f(x)+ay=f(x)+ay=f(x)+a

Vertical stretches and reflections: y=af(x)y=af(x)y=af(x)

Horizontal translations: y=f(x−a)y=f(x-a)y=f(x−a)

Horizontal stretches and compressions: y=f(kx)y=f(kx)y=f(kx)

Reflections in the axes

Reflection in the xxx-axis: y=−f(x)y=-f(x)y=−f(x)

Reflection in the yyy-axis: y=f(−x)y=f(-x)y=f(−x)

Combining transformations

Describing transformations in words

Recap questions

You've reached the end

Transforming Graphs y=f(x) Revision Guide

The starting point: $y = f (x)$

Vertical translations: $y = f (x) + a$

Vertical stretches and reflections: $y = a f (x)$

Horizontal translations: $y = f (x - a)$

Horizontal stretches and compressions: $y = f (k x)$

Reflection in the $x$ -axis: $y = - f (x)$

Reflection in the $y$ -axis: $y = f (- x)$

The starting point: $y = f (x)$

Vertical translations: $y = f (x) + a$

Vertical stretches and reflections: $y = a f (x)$

Horizontal translations: $y = f (x - a)$

Horizontal stretches and compressions: $y = f (k x)$

Reflection in the $x$ -axis: $y = - f (x)$

Reflection in the $y$ -axis: $y = f (- x)$