Transforming Graphs y=f(x)

What you'll learn

How to treat $y = f (x)$ as the original graph.
How outside changes affect $y$ -coordinates directly.
How inside-the-bracket changes affect $x$ -coordinates in the opposite way.
How to transform key points when graphs are shifted, stretched or reflected.

1. Start with the original graph

Before transforming anything, make sure you know what the notation means.

Definition

Key language

A function is a rule that turns an input $x$ into an output $f (x)$ .
The graph of $y = f (x)$ is the original graph.
A transformation is a change to a graph, such as a slide, stretch or reflection.
A key point is a useful point to track, such as a turning point, where a curve changes direction, or an intercept, where it crosses an axis.

If a point $(a, b)$ lies on $y = f (x)$ , then $f (a) = b$ . This is the link between coordinates and function notation.

Example

Reading a point from function notation

Suppose you are told that $f (- 2) = 5$ .
The input is -2, so the $x$ -coordinate is -2.
The output is 5, so the $y$ -coordinate is 5.
Therefore the original graph $y = f (x)$ passes through $(- 2, 5)$ .

The statement f(-2)=5 means the graph of y=f(x) contains the point (-2,5).

Key Idea

Move points, then draw the curve

To sketch a transformed graph, move several key points first. Then join them with the same general curve shape.

2. Changes outside $f (x)$ : vertical movement

A change outside the function happens after the function has produced the output. So it changes the $y$ -coordinates.

A vertical change means up or down, parallel to the $y$ -axis.

For $y = f (x) + a$ , every point moves by adding $a$ to its $y$ -coordinate:

positive $a$ means up
negative $a$ means down

A translation is a slide: every point moves the same distance in the same direction.

Example

Vertical translation: $y = f (x) - 3$

Suppose the original graph has key points $(1, 4)$ and $(- 2, 0)$ .
The -3 is outside $f (x)$ , so change only the $y$ -coordinates.
Subtract 3 from each $y$ -coordinate: $(1, 4)$ becomes $(1, 1)$ , and $(- 2, 0)$ becomes $(- 2, - 3)$ .
The whole graph has moved 3 units down.

A vertical translation by -3 keeps x-coordinates the same and moves every point 3 units down.

Common Mistake

Changing the wrong coordinate

If the change is outside $f (x)$ , do not change the $x$ -coordinates. Outside changes outputs, so it changes $y$ .

3. Multiplying outside: vertical stretches and reflections

A stretch changes distances from an axis by a scale factor, which is the multiplier used.

For $y = k f (x)$ , every $y$ -coordinate is multiplied by $k$ :

if $k > 1$ , the graph is stretched vertically
if $0 < k < 1$ , the graph is squashed vertically
if $k < 0$ , the graph is reflected in the $x$ -axis as well

A reflection is a mirror image in a line.

Example

Vertical stretch and reflection: $y = - 2 f (x)$

Suppose $y = f (x)$ passes through $(- 3, 1)$ , $(0, - 2)$ and $(2, 4)$ .
The multiplier -2 is outside $f (x)$ , so keep the $x$ -coordinates the same.
Multiply each $y$ -coordinate by -2: $(- 3, 1)$ becomes $(- 3, - 2)$ , $(0, - 2)$ becomes $(0, 4)$ , and $(2, 4)$ becomes $(2, - 8)$ .
The graph is reflected in the $x$ -axis and stretched vertically by scale factor 2.

Multiplying outside by -2 keeps x-coordinates fixed, doubles distances from the x-axis and reflects in the x-axis.

Common Mistake

Reflection axis mix-up

For $y = - f (x)$ , the $y$ -coordinates change sign but the $x$ -coordinates stay the same, so the mirror line is the $x$ -axis.

4. Changes inside the bracket: horizontal movement

A change inside the bracket changes the input before the function acts. So it affects the $x$ -coordinates.

A horizontal change means left or right, parallel to the $x$ -axis.

The important rule is:

Inside changes work the opposite way for $x$ -coordinates.

For example:

$y = f (x - 5)$ moves the graph right 5
$y = f (x + 4)$ moves the graph left 4

Why? Because the expression inside the bracket must equal the old input.

Example

Horizontal translation: $y = f (x - 4)$

Suppose the original graph has a turning point at $(- 1, 2)$ and another point at $(3, 5)$ .
The $x - 4$ is inside the bracket, so change only the $x$ -coordinates.
Inside changes work oppositely, so add 4 to each $x$ -coordinate.
The turning point $(- 1, 2)$ becomes $(3, 2)$ , and the point $(3, 5)$ becomes $(7, 5)$ .
The whole graph moves 4 units to the right.

For y=f(x-4), every x-coordinate increases by 4 while y-coordinates stay the same.

Common Mistake

Forgetting the opposite sign

$f (x + 4)$ moves the graph left 4, not right 4. Inside the bracket, $x$ does the opposite.

5. Multiplying inside: horizontal stretches and reflections

For $y = f (k x)$ , the multiplier is inside the bracket, so it affects the $x$ -coordinates.

To find the new $x$ -coordinate, divide the old $x$ -coordinate by $k$ .

So:

$y = f (2 x)$ halves the $x$ -coordinates
$y = f (5 x)$ divides the $x$ -coordinates by 5
$y = f (- x)$ reflects the graph in the $y$ -axis

Example

Horizontal stretch and reflection: $y = f (- 2 x)$

Suppose the original graph contains the points $(- 4, 3)$ , $(0, - 1)$ and $(6, 2)$ .
The -2 is inside the bracket, so the $y$ -coordinates stay the same.
Divide each old $x$ -coordinate by -2: $(- 4, 3)$ becomes $(2, 3)$ , $(0, - 1)$ becomes $(0, - 1)$ , and $(6, 2)$ becomes $(- 3, 2)$ .
The graph is reflected in the $y$ -axis and compressed horizontally by scale factor $12\frac{1}{2}$ .

For y=f(-2x), y-coordinates stay the same while x-coordinates are divided by -2.

Tip

Quick memory

Outside affects outputs, so $y$ changes directly. Inside affects inputs, so solve the bracket equal to the old $x$ .

6. Combining transformations

For harder questions, use a point map, which is a rule showing where each old point moves.

If an old point is $(p, q)$ on $y = f (x)$ , then for a graph like $y = a f (b x + c) + d$ :

solve $b x + c = p$ to get the new $x$ -coordinate
calculate $a q + d$ to get the new $y$ -coordinate

This avoids guessing the order of transformations.

Example

Combined transformation: $y = 3 f (2 x - 4) + 5$

Suppose the original graph has a turning point at $(2, - 1)$ .
The old input is $p = 2$ , so solve $2 x - 4 = 2$ .
This gives $2 x = 6$ , so the new $x$ -coordinate is 3.
The old output is $q = - 1$ , so the new $y$ -coordinate is $3 (- 1) + 5 = 2$ .
The new turning point is $(3, 2)$ .

The point map for y=3f(2x-4)+5 sends the old turning point (2,-1) to (3,2).

7. Working backwards from a graph

Sometimes you are shown the original graph and the transformed graph, and you must write the equation.

Compare matching key points. Look at what happened to the $x$ -coordinates and $y$ -coordinates separately.

Example

Writing the equation from a shifted graph

Suppose the original curve has a vertex at $(0, 0)$ .
The new curve has the same shape, with its vertex at $(5, - 2)$ .
The $x$ -coordinate has increased by 5, so the inside change is $x - 5$ .
The $y$ -coordinate has decreased by 2, so the outside change is -2.
The transformed graph is $y = f (x - 5) - 2$ .

A vertex moving from (0,0) to (5,-2) shows a translation 5 units right and 2 units down.

Exam technique

In the exam

Label two or three key points on the original graph before transforming anything.
Decide whether each change is outside $f$ or inside the bracket.
For inside changes, remember the opposite effect on $x$ , or solve the bracket equal to the old input.

Self review

Check yourself

If $(2, 5)$ lies on $y = f (x)$ , where does it go on $y = f (x + 3)$ ?
Which axis is the mirror line for $y = - f (x)$ , and which for $y = f (- x)$ ?
For $y = 3 f (2 x) - 1$ , what happens to the $x$ -coordinates and $y$ -coordinates?

Transforming Graphs y=f(x)

What you'll learn

How to treat $y = f (x)$ as the original graph.
How outside changes affect $y$ -coordinates directly.
How inside-the-bracket changes affect $x$ -coordinates in the opposite way.
How to transform key points when graphs are shifted, stretched or reflected.

1. Start with the original graph

Before transforming anything, make sure you know what the notation means.

Definition

Key language

A function is a rule that turns an input $x$ into an output $f (x)$ .
The graph of $y = f (x)$ is the original graph.
A transformation is a change to a graph, such as a slide, stretch or reflection.
A key point is a useful point to track, such as a turning point, where a curve changes direction, or an intercept, where it crosses an axis.

If a point $(a, b)$ lies on $y = f (x)$ , then $f (a) = b$ . This is the link between coordinates and function notation.

Example

Reading a point from function notation

Suppose you are told that $f (- 2) = 5$ .
The input is -2, so the $x$ -coordinate is -2.
The output is 5, so the $y$ -coordinate is 5.
Therefore the original graph $y = f (x)$ passes through $(- 2, 5)$ .

The statement f(-2)=5 means the graph of y=f(x) contains the point (-2,5).

Key Idea

Move points, then draw the curve

To sketch a transformed graph, move several key points first. Then join them with the same general curve shape.

2. Changes outside $f (x)$ : vertical movement

A change outside the function happens after the function has produced the output. So it changes the $y$ -coordinates.

A vertical change means up or down, parallel to the $y$ -axis.

For $y = f (x) + a$ , every point moves by adding $a$ to its $y$ -coordinate:

positive $a$ means up
negative $a$ means down

A translation is a slide: every point moves the same distance in the same direction.

Example

Vertical translation: $y = f (x) - 3$

Suppose the original graph has key points $(1, 4)$ and $(- 2, 0)$ .
The -3 is outside $f (x)$ , so change only the $y$ -coordinates.
Subtract 3 from each $y$ -coordinate: $(1, 4)$ becomes $(1, 1)$ , and $(- 2, 0)$ becomes $(- 2, - 3)$ .
The whole graph has moved 3 units down.

A vertical translation by -3 keeps x-coordinates the same and moves every point 3 units down.

Common Mistake

Changing the wrong coordinate

If the change is outside $f (x)$ , do not change the $x$ -coordinates. Outside changes outputs, so it changes $y$ .

3. Multiplying outside: vertical stretches and reflections

A stretch changes distances from an axis by a scale factor, which is the multiplier used.

For $y = k f (x)$ , every $y$ -coordinate is multiplied by $k$ :

if $k > 1$ , the graph is stretched vertically
if $0 < k < 1$ , the graph is squashed vertically
if $k < 0$ , the graph is reflected in the $x$ -axis as well

A reflection is a mirror image in a line.

Example

Vertical stretch and reflection: $y = - 2 f (x)$

Suppose $y = f (x)$ passes through $(- 3, 1)$ , $(0, - 2)$ and $(2, 4)$ .
The multiplier -2 is outside $f (x)$ , so keep the $x$ -coordinates the same.
Multiply each $y$ -coordinate by -2: $(- 3, 1)$ becomes $(- 3, - 2)$ , $(0, - 2)$ becomes $(0, 4)$ , and $(2, 4)$ becomes $(2, - 8)$ .
The graph is reflected in the $x$ -axis and stretched vertically by scale factor 2.

Multiplying outside by -2 keeps x-coordinates fixed, doubles distances from the x-axis and reflects in the x-axis.

Common Mistake

Reflection axis mix-up

For $y = - f (x)$ , the $y$ -coordinates change sign but the $x$ -coordinates stay the same, so the mirror line is the $x$ -axis.

4. Changes inside the bracket: horizontal movement

A change inside the bracket changes the input before the function acts. So it affects the $x$ -coordinates.

A horizontal change means left or right, parallel to the $x$ -axis.

The important rule is:

Inside changes work the opposite way for $x$ -coordinates.

For example:

$y = f (x - 5)$ moves the graph right 5
$y = f (x + 4)$ moves the graph left 4

Why? Because the expression inside the bracket must equal the old input.

Example

Horizontal translation: $y = f (x - 4)$

Suppose the original graph has a turning point at $(- 1, 2)$ and another point at $(3, 5)$ .
The $x - 4$ is inside the bracket, so change only the $x$ -coordinates.
Inside changes work oppositely, so add 4 to each $x$ -coordinate.
The turning point $(- 1, 2)$ becomes $(3, 2)$ , and the point $(3, 5)$ becomes $(7, 5)$ .
The whole graph moves 4 units to the right.

For y=f(x-4), every x-coordinate increases by 4 while y-coordinates stay the same.

Common Mistake

Forgetting the opposite sign

$f (x + 4)$ moves the graph left 4, not right 4. Inside the bracket, $x$ does the opposite.

5. Multiplying inside: horizontal stretches and reflections

For $y = f (k x)$ , the multiplier is inside the bracket, so it affects the $x$ -coordinates.

To find the new $x$ -coordinate, divide the old $x$ -coordinate by $k$ .

So:

$y = f (2 x)$ halves the $x$ -coordinates
$y = f (5 x)$ divides the $x$ -coordinates by 5
$y = f (- x)$ reflects the graph in the $y$ -axis

Example

Horizontal stretch and reflection: $y = f (- 2 x)$

Suppose the original graph contains the points $(- 4, 3)$ , $(0, - 1)$ and $(6, 2)$ .
The -2 is inside the bracket, so the $y$ -coordinates stay the same.
Divide each old $x$ -coordinate by -2: $(- 4, 3)$ becomes $(2, 3)$ , $(0, - 1)$ becomes $(0, - 1)$ , and $(6, 2)$ becomes $(- 3, 2)$ .
The graph is reflected in the $y$ -axis and compressed horizontally by scale factor $12\frac{1}{2}$ .

For y=f(-2x), y-coordinates stay the same while x-coordinates are divided by -2.

Tip

Quick memory

Outside affects outputs, so $y$ changes directly. Inside affects inputs, so solve the bracket equal to the old $x$ .

6. Combining transformations

For harder questions, use a point map, which is a rule showing where each old point moves.

If an old point is $(p, q)$ on $y = f (x)$ , then for a graph like $y = a f (b x + c) + d$ :

solve $b x + c = p$ to get the new $x$ -coordinate
calculate $a q + d$ to get the new $y$ -coordinate

This avoids guessing the order of transformations.

Example

Combined transformation: $y = 3 f (2 x - 4) + 5$

Suppose the original graph has a turning point at $(2, - 1)$ .
The old input is $p = 2$ , so solve $2 x - 4 = 2$ .
This gives $2 x = 6$ , so the new $x$ -coordinate is 3.
The old output is $q = - 1$ , so the new $y$ -coordinate is $3 (- 1) + 5 = 2$ .
The new turning point is $(3, 2)$ .

The point map for y=3f(2x-4)+5 sends the old turning point (2,-1) to (3,2).

7. Working backwards from a graph

Sometimes you are shown the original graph and the transformed graph, and you must write the equation.

Compare matching key points. Look at what happened to the $x$ -coordinates and $y$ -coordinates separately.

Example

Writing the equation from a shifted graph

Suppose the original curve has a vertex at $(0, 0)$ .
The new curve has the same shape, with its vertex at $(5, - 2)$ .
The $x$ -coordinate has increased by 5, so the inside change is $x - 5$ .
The $y$ -coordinate has decreased by 2, so the outside change is -2.
The transformed graph is $y = f (x - 5) - 2$ .

A vertex moving from (0,0) to (5,-2) shows a translation 5 units right and 2 units down.

Exam technique

In the exam

Label two or three key points on the original graph before transforming anything.
Decide whether each change is outside $f$ or inside the bracket.
For inside changes, remember the opposite effect on $x$ , or solve the bracket equal to the old input.

Self review

Check yourself

If $(2, 5)$ lies on $y = f (x)$ , where does it go on $y = f (x + 3)$ ?
Which axis is the mirror line for $y = - f (x)$ , and which for $y = f (- x)$ ?
For $y = 3 f (2 x) - 1$ , what happens to the $x$ -coordinates and $y$ -coordinates?

Transforming Graphs y=f(x)

What you'll learn

1. Start with the original graph

2. Changes outside $f (x)$ : vertical movement

3. Multiplying outside: vertical stretches and reflections

4. Changes inside the bracket: horizontal movement

5. Multiplying inside: horizontal stretches and reflections

6. Combining transformations

7. Working backwards from a graph

Recap questions

You've reached the end

Transforming Graphs y=f(x)

What you'll learn

1. Start with the original graph

2. Changes outside $f (x)$ : vertical movement

3. Multiplying outside: vertical stretches and reflections

4. Changes inside the bracket: horizontal movement

5. Multiplying inside: horizontal stretches and reflections

6. Combining transformations

7. Working backwards from a graph

Recap questions

You've reached the end

Transforming Graphs y=f(x)

What you'll learn

1. Start with the original graph

2. Changes outside f(x)f(x)f(x): vertical movement

3. Multiplying outside: vertical stretches and reflections

4. Changes inside the bracket: horizontal movement

5. Multiplying inside: horizontal stretches and reflections

6. Combining transformations

7. Working backwards from a graph

Recap questions

You've reached the end

Transforming Graphs y=f(x) Revision Guide

Transforming Graphs y=f(x)

What you'll learn

1. Start with the original graph

2. Changes outside f(x)f(x)f(x): vertical movement

3. Multiplying outside: vertical stretches and reflections

4. Changes inside the bracket: horizontal movement

5. Multiplying inside: horizontal stretches and reflections

6. Combining transformations

7. Working backwards from a graph

Recap questions

You've reached the end

Transforming Graphs y=f(x) Revision Guide

2. Changes outside $f (x)$ : vertical movement

2. Changes outside $f (x)$ : vertical movement