1.2.23 Simple graph transformations

What you'll learn

How to recognise transformations of the graph $y = f (x)$ .
How to sketch $y = a f (x)$ , $y = f (x) + a$ , $y = f (x + a)$ and $y = f (a x)$ .
How to describe transformations precisely using translations, stretches and reflections.
How to track key points and find equations of transformed graphs.

Starting point: the graph of a function

The notation $y = f (x)$ means that the function $f$ takes an input $x$ and produces an output $y$ .

A graph transformation changes the position or shape of a graph according to a rule. The main transformations are:

a translation, which moves the graph without changing its shape;
a stretch or compression, which changes distances in one direction;
a reflection, which flips the graph in a line.

To sketch a transformed graph accurately, it helps to track important points such as intercepts, turning points and endpoints.

Definition

Point mapping

A point mapping describes where each point moves. If $(p, q)$ lies on $y = f (x)$ , then $q = f (p)$ . A transformation changes $(p, q)$ to a new point according to its rule.

Changes outside the function

In $y = a f (x)$ and $y = f (x) + a$ , the change is made outside the function. It therefore acts directly on the output, or $y$ -coordinate.

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

Adding $a$ to the output moves every point vertically:

(p,q)⟼(p,q+a).(p,q)\longmapsto(p,q+a).

Therefore, $y = f (x) + a$ is a translation of $y = f (x)$ :

upwards by $a$ units if $a > 0$ ;
downwards by $∣ a ∣$ units if $a < 0$ .

The translation vector is

(0a).\begin{pmatrix} 0\\ a \end{pmatrix}.

Key Idea

Outside addition

Adding a number outside the function causes a vertical movement in the same direction as the sign: $+ a$ moves up when $a > 0$ .

Example

Translating a quadratic vertically

Sketch $y=x^2-4$ from the graph of $y=x^2$ .

Write the equation as $y = f (x) - 4$ , where $f(x)=x^2$ . This is an outside change, so it is vertical.
Map every point using $(p,q)↦(p,q−4)(p,q)\mapsto(p,q-4)$ , which moves the graph down 4 units.
The original turning point $(0, 0)$ moves to $(0, - 4)$ , so the transformed parabola has turning point $(0, - 4)$ and the same shape as $y=x^2$ .

Vertical scaling and reflection: $y = a f (x)$

Multiplying the entire output by $a$ changes every $y$ -coordinate:

(p,q)⟼(p,aq).(p,q)\longmapsto(p,aq).

For $a≠0a\neq 0$ , this is a stretch parallel to the $y$ -axis with scale factor $∣ a ∣$ . In particular:

if $∣ a ∣ > 1$ , the graph is stretched vertically;
if $0 < ∣ a ∣ < 1$ , the graph is compressed vertically;
if $a < 0$ , the graph is also reflected in the $x$ -axis;
if $a = 0$ , the equation becomes $y = 0$ , so the graph is the $x$ -axis wherever the original function is defined.

Points on the $x$ -axis do not move because multiplying a zero $y$ -coordinate still gives zero. Therefore, the roots usually remain unchanged.

Outside graph transformations showing vertical stretches, a reflection in the x-axis and an upward translation

Example

Stretching and reflecting a quadratic

Describe the transformation from $y=x^2-1$ to $y=-3(x^2-1)$ , and identify the new turning point.

Let $f(x)=x^2-1$ . The new equation is $y = - 3 f (x)$ , so every $y$ -coordinate is multiplied by $- 3$ .
Since $∣ - 3∣ = 3$ , there is a stretch parallel to the $y$ -axis with scale factor 3. The negative sign also gives a reflection in the $x$ -axis.
The original turning point is $(0, - 1)$ . Using $(p,q)↦(p,−3q)(p,q)\mapsto(p,-3q)$ gives $(0,−1)↦(0,3)(0,-1)\mapsto(0,3)$ .

Common Mistake

Stretching both coordinates

For $y = a f (x)$ , multiply only the $y$ -coordinates by $a$ . The $x$ -coordinates do not change.

Changes inside the function

In $y = f (x + a)$ and $y = f (a x)$ , the change is made to the input. It therefore affects the $x$ -coordinates.

Inside transformations appear to work in the opposite direction because you must compensate for the altered input.

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

Suppose $(p, q)$ is on the original graph, so $f (p) = q$ . On the new graph $y = f (x + a)$ , the same output occurs when

x + a = p,

so $x = p - a$ . Therefore,

(p,q)⟼(p−a,q).(p,q)\longmapsto(p-a,q).

Hence $y = f (x + a)$ is a translation by the vector

(−a0).\begin{pmatrix} -a\\ 0 \end{pmatrix}.

For example, $y = f (x + 3)$ moves left 3 units, whereas $y = f (x - 3)$ moves right 3 units.

Key Idea

Inside addition is opposite

For horizontal translations, the direction is opposite to the sign inside the brackets: $f (x + a)$ moves left when $a > 0$ .

Example

Translating a square-root graph

Find the equation obtained by translating $y=xy=\sqrt{x}$ right by 5 units.

A movement right by 5 changes each $x$ -coordinate according to $(p,q)↦(p+5,q)(p,q)\mapsto(p+5,q)$ .
An inside subtraction produces a movement to the right, so replace $x$ with $x - 5$ .
The transformed equation is $y=x−5y=\sqrt{x-5}$ . Its endpoint moves from $(0, 0)$ to $(5, 0)$ .

Horizontal scaling and reflection: $y = f (a x)$

To obtain the same output $q = f (p)$ from $y = f (a x)$ , the input must satisfy

a x = p .

When $a≠0a\neq 0$ , this gives $x=pax=\frac{p}{a}$ , so

(p,q)⟼(pa,q).(p,q)\longmapsto\left(\frac{p}{a},q\right).

The graph undergoes a stretch parallel to the $x$ -axis with scale factor $1∣a∣\frac{1}{|a|}$ :

if $∣ a ∣ > 1$ , the graph is compressed horizontally;
if $0 < ∣ a ∣ < 1$ , the graph is stretched horizontally;
if $a < 0$ , the graph is also reflected in the $y$ -axis.

Inside graph transformations showing horizontal translations, a horizontal compression and a reflection in the y-axis

Example

Compressing a sine graph horizontally

Describe the transformation from $y=sin⁡xy=\sin x$ to $y=sin⁡(2x)y=\sin(2x)$ and find where the point $(π2,1)\left(\frac{\pi}{2},1\right)$ moves.

The multiplier 2 is inside the function, so it acts on the $x$ -coordinates.
The horizontal scale factor is $12\frac{1}{2}$ , meaning that all horizontal distances are halved.
Apply the mapping $(p,q)↦(p2,q)(p,q)\mapsto\left(\frac{p}{2},q\right)$ :

(π2,1)⟼(π4,1).\left(\frac{\pi}{2},1\right)\longmapsto\left(\frac{\pi}{4},1\right).

Common Mistake

Using the visible multiplier

The graph of $y = f (3 x)$ has horizontal scale factor $13\frac{1}{3}$ , not 3. Inside multipliers act reciprocally on $x$ -coordinates.

Common Mistake

The case a = 0

The reciprocal rule for $y = f (a x)$ requires $a≠0a\neq 0$ . If $a = 0$ , then $y = f (0)$ is a constant horizontal line, provided that $f (0)$ is defined.

Finding equations from descriptions

Translate each verbal description into the part of the function that must change:

vertical movement: add outside $f (x)$ ;
horizontal movement: add or subtract inside the input;
vertical scaling: multiply outside $f (x)$ ;
horizontal scaling: multiply the input.

Example

Combining a translation and a stretch

The graph $y = f (x)$ is stretched parallel to the $y$ -axis with scale factor 2, then translated left by 3 units. Find its new equation.

The vertical stretch multiplies every output by 2, giving $y = 2 f (x)$ .
Translating left by 3 requires replacing the input $x$ with $x + 3$ .
Applying both changes gives

y = 2 f (x + 3) .

A point $(p, q)$ on the original graph moves to $(p - 3, 2 q)$ .

Tip

Outside and inside

Changes outside $f$ affect height and therefore act on $y$ -coordinates. Changes inside the brackets affect position across the page and therefore act reciprocally on $x$ -coordinates.

Describing transformations precisely

State all of the following where relevant:

the type of transformation;
the direction or line of reflection;
the scale factor or translation vector.

For example, describe $y = f (- 2 x)$ as a reflection in the $y$ -axis together with a stretch parallel to the $x$ -axis with scale factor $12\frac{1}{2}$ .

Exam technique

In the exam

Decide whether the change is inside or outside the function before moving any points.
Mark transformed intercepts, turning points and endpoints first, then join them with the correct general shape.
Check the direction of horizontal translations and use reciprocal scale factors for horizontal stretches.
When describing a transformation, include the axis direction, scale factor, reflection line or translation vector as appropriate.

Self review

Check yourself

Where does the point $(4, - 2)$ move under the transformation $y = 3 f (x) + 1$ ?
How would you transform $y = f (x)$ to obtain $y = f (x - 6)$ ?
Describe fully the transformation from $y = f (x)$ to $y=f(−12x)y=f(-\frac{1}{2}x)$ .

What you'll learn

How to recognise transformations of the graph $y = f (x)$ .
How to sketch $y = a f (x)$ , $y = f (x) + a$ , $y = f (x + a)$ and $y = f (a x)$ .
How to describe transformations precisely using translations, stretches and reflections.
How to track key points and find equations of transformed graphs.

Starting point: the graph of a function

The notation $y = f (x)$ means that the function $f$ takes an input $x$ and produces an output $y$ .

A graph transformation changes the position or shape of a graph according to a rule. The main transformations are:

a translation, which moves the graph without changing its shape;
a stretch or compression, which changes distances in one direction;
a reflection, which flips the graph in a line.

To sketch a transformed graph accurately, it helps to track important points such as intercepts, turning points and endpoints.

Definition

Point mapping

A point mapping describes where each point moves. If $(p, q)$ lies on $y = f (x)$ , then $q = f (p)$ . A transformation changes $(p, q)$ to a new point according to its rule.

Changes outside the function

In $y = a f (x)$ and $y = f (x) + a$ , the change is made outside the function. It therefore acts directly on the output, or $y$ -coordinate.

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

Adding $a$ to the output moves every point vertically:

(p,q)⟼(p,q+a).(p,q)\longmapsto(p,q+a).

Therefore, $y = f (x) + a$ is a translation of $y = f (x)$ :

upwards by $a$ units if $a > 0$ ;
downwards by $∣ a ∣$ units if $a < 0$ .

The translation vector is

(0a).\begin{pmatrix} 0\\ a \end{pmatrix}.

Key Idea

Outside addition

Adding a number outside the function causes a vertical movement in the same direction as the sign: $+ a$ moves up when $a > 0$ .

Example

Translating a quadratic vertically

Sketch $y=x^2-4$ from the graph of $y=x^2$ .

Write the equation as $y = f (x) - 4$ , where $f(x)=x^2$ . This is an outside change, so it is vertical.
Map every point using $(p,q)↦(p,q−4)(p,q)\mapsto(p,q-4)$ , which moves the graph down 4 units.
The original turning point $(0, 0)$ moves to $(0, - 4)$ , so the transformed parabola has turning point $(0, - 4)$ and the same shape as $y=x^2$ .

Vertical scaling and reflection: $y = a f (x)$

Multiplying the entire output by $a$ changes every $y$ -coordinate:

(p,q)⟼(p,aq).(p,q)\longmapsto(p,aq).

For $a≠0a\neq 0$ , this is a stretch parallel to the $y$ -axis with scale factor $∣ a ∣$ . In particular:

if $∣ a ∣ > 1$ , the graph is stretched vertically;
if $0 < ∣ a ∣ < 1$ , the graph is compressed vertically;
if $a < 0$ , the graph is also reflected in the $x$ -axis;
if $a = 0$ , the equation becomes $y = 0$ , so the graph is the $x$ -axis wherever the original function is defined.

Points on the $x$ -axis do not move because multiplying a zero $y$ -coordinate still gives zero. Therefore, the roots usually remain unchanged.

Outside graph transformations showing vertical stretches, a reflection in the x-axis and an upward translation

Example

Stretching and reflecting a quadratic

Describe the transformation from $y=x^2-1$ to $y=-3(x^2-1)$ , and identify the new turning point.

Let $f(x)=x^2-1$ . The new equation is $y = - 3 f (x)$ , so every $y$ -coordinate is multiplied by $- 3$ .
Since $∣ - 3∣ = 3$ , there is a stretch parallel to the $y$ -axis with scale factor 3. The negative sign also gives a reflection in the $x$ -axis.
The original turning point is $(0, - 1)$ . Using $(p,q)↦(p,−3q)(p,q)\mapsto(p,-3q)$ gives $(0,−1)↦(0,3)(0,-1)\mapsto(0,3)$ .

Common Mistake

Stretching both coordinates

For $y = a f (x)$ , multiply only the $y$ -coordinates by $a$ . The $x$ -coordinates do not change.

Changes inside the function

In $y = f (x + a)$ and $y = f (a x)$ , the change is made to the input. It therefore affects the $x$ -coordinates.

Inside transformations appear to work in the opposite direction because you must compensate for the altered input.

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

Suppose $(p, q)$ is on the original graph, so $f (p) = q$ . On the new graph $y = f (x + a)$ , the same output occurs when

x + a = p,

so $x = p - a$ . Therefore,

(p,q)⟼(p−a,q).(p,q)\longmapsto(p-a,q).

Hence $y = f (x + a)$ is a translation by the vector

(−a0).\begin{pmatrix} -a\\ 0 \end{pmatrix}.

For example, $y = f (x + 3)$ moves left 3 units, whereas $y = f (x - 3)$ moves right 3 units.

Key Idea

Inside addition is opposite

For horizontal translations, the direction is opposite to the sign inside the brackets: $f (x + a)$ moves left when $a > 0$ .

Example

Translating a square-root graph

Find the equation obtained by translating $y=xy=\sqrt{x}$ right by 5 units.

A movement right by 5 changes each $x$ -coordinate according to $(p,q)↦(p+5,q)(p,q)\mapsto(p+5,q)$ .
An inside subtraction produces a movement to the right, so replace $x$ with $x - 5$ .
The transformed equation is $y=x−5y=\sqrt{x-5}$ . Its endpoint moves from $(0, 0)$ to $(5, 0)$ .

Horizontal scaling and reflection: $y = f (a x)$

To obtain the same output $q = f (p)$ from $y = f (a x)$ , the input must satisfy

a x = p .

When $a≠0a\neq 0$ , this gives $x=pax=\frac{p}{a}$ , so

(p,q)⟼(pa,q).(p,q)\longmapsto\left(\frac{p}{a},q\right).

The graph undergoes a stretch parallel to the $x$ -axis with scale factor $1∣a∣\frac{1}{|a|}$ :

if $∣ a ∣ > 1$ , the graph is compressed horizontally;
if $0 < ∣ a ∣ < 1$ , the graph is stretched horizontally;
if $a < 0$ , the graph is also reflected in the $y$ -axis.

Inside graph transformations showing horizontal translations, a horizontal compression and a reflection in the y-axis

Example

Compressing a sine graph horizontally

Describe the transformation from $y=sin⁡xy=\sin x$ to $y=sin⁡(2x)y=\sin(2x)$ and find where the point $(π2,1)\left(\frac{\pi}{2},1\right)$ moves.

The multiplier 2 is inside the function, so it acts on the $x$ -coordinates.
The horizontal scale factor is $12\frac{1}{2}$ , meaning that all horizontal distances are halved.
Apply the mapping $(p,q)↦(p2,q)(p,q)\mapsto\left(\frac{p}{2},q\right)$ :

(π2,1)⟼(π4,1).\left(\frac{\pi}{2},1\right)\longmapsto\left(\frac{\pi}{4},1\right).

Common Mistake

Using the visible multiplier

The graph of $y = f (3 x)$ has horizontal scale factor $13\frac{1}{3}$ , not 3. Inside multipliers act reciprocally on $x$ -coordinates.

Common Mistake

The case a = 0

The reciprocal rule for $y = f (a x)$ requires $a≠0a\neq 0$ . If $a = 0$ , then $y = f (0)$ is a constant horizontal line, provided that $f (0)$ is defined.

Finding equations from descriptions

Translate each verbal description into the part of the function that must change:

vertical movement: add outside $f (x)$ ;
horizontal movement: add or subtract inside the input;
vertical scaling: multiply outside $f (x)$ ;
horizontal scaling: multiply the input.

Example

Combining a translation and a stretch

The graph $y = f (x)$ is stretched parallel to the $y$ -axis with scale factor 2, then translated left by 3 units. Find its new equation.

The vertical stretch multiplies every output by 2, giving $y = 2 f (x)$ .
Translating left by 3 requires replacing the input $x$ with $x + 3$ .
Applying both changes gives

y = 2 f (x + 3) .

A point $(p, q)$ on the original graph moves to $(p - 3, 2 q)$ .

Tip

Outside and inside

Changes outside $f$ affect height and therefore act on $y$ -coordinates. Changes inside the brackets affect position across the page and therefore act reciprocally on $x$ -coordinates.

Describing transformations precisely

State all of the following where relevant:

the type of transformation;
the direction or line of reflection;
the scale factor or translation vector.

For example, describe $y = f (- 2 x)$ as a reflection in the $y$ -axis together with a stretch parallel to the $x$ -axis with scale factor $12\frac{1}{2}$ .

Exam technique

In the exam

Decide whether the change is inside or outside the function before moving any points.
Mark transformed intercepts, turning points and endpoints first, then join them with the correct general shape.
Check the direction of horizontal translations and use reciprocal scale factors for horizontal stretches.
When describing a transformation, include the axis direction, scale factor, reflection line or translation vector as appropriate.

Self review

Check yourself

Where does the point $(4, - 2)$ move under the transformation $y = 3 f (x) + 1$ ?
How would you transform $y = f (x)$ to obtain $y = f (x - 6)$ ?
Describe fully the transformation from $y = f (x)$ to $y=f(−12x)y=f(-\frac{1}{2}x)$ .

What you'll learn

Starting point: the graph of a function

Changes outside the function

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

Vertical scaling and reflection: $y = a f (x)$

Changes inside the function

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

Horizontal scaling and reflection: $y = f (a x)$

Finding equations from descriptions

Describing transformations precisely

Review 1.2.23 Simple graph transformations by teaching Genie

1.2.23 Simple graph transformations

What you'll learn

Starting point: the graph of a function

Changes outside the function

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

Vertical scaling and reflection: $y = a f (x)$

Changes inside the function

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

Horizontal scaling and reflection: $y = f (a x)$

Finding equations from descriptions

Describing transformations precisely

Review 1.2.23 Simple graph transformations by teaching Genie

1.2.23 Simple graph transformations

What you'll learn

Starting point: the graph of a function

Changes outside the function

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

Vertical scaling and reflection: y=af(x)y=af(x)y=af(x)

Changes inside the function

Horizontal translations: y=f(x+a)y=f(x+a)y=f(x+a)

Horizontal scaling and reflection: y=f(ax)y=f(ax)y=f(ax)

Finding equations from descriptions

Describing transformations precisely

Review 1.2.23 Simple graph transformations by teaching Genie

1.2.23 Simple graph transformations Revision Guide

1.2.23 Simple graph transformations

What you'll learn

Starting point: the graph of a function

Changes outside the function

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

Vertical scaling and reflection: y=af(x)y=af(x)y=af(x)

Changes inside the function

Horizontal translations: y=f(x+a)y=f(x+a)y=f(x+a)

Horizontal scaling and reflection: y=f(ax)y=f(ax)y=f(ax)

Finding equations from descriptions

Describing transformations precisely

Review 1.2.23 Simple graph transformations by teaching Genie

1.2.23 Simple graph transformations Revision Guide

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

Vertical scaling and reflection: $y = a f (x)$

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

Horizontal scaling and reflection: $y = f (a x)$

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

Vertical scaling and reflection: $y = a f (x)$

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

Horizontal scaling and reflection: $y = f (a x)$