Graphs and Transformations

What you'll learn

How to sketch polynomial and reciprocal graphs from their equations.
How to find and label intercepts, roots, turning points and asymptotes.
How transformations such as $f (x) + a$ , $f (x - a)$ , $- f (x)$ and $f (a x)$ change a graph.
How to use a sketch to solve equations and inequalities.

1. Reading a graph: the essential vocabulary

A graph is a picture of all the points $(x, y)$ that satisfy an equation, such as $y = f (x)$ .

Definition

Intercepts, roots and asymptotes

An x-intercept is where a graph crosses or touches the x-axis, so $y = 0$ .
A y-intercept is where a graph crosses the y-axis, so $x = 0$ .
A root or zero of $f (x)$ is a value of $x$ where $f (x) = 0$ .
An asymptote is a straight line that a curve gets closer and closer to but does not meet.

For factorised polynomial graphs, the roots are usually the easiest part. If

f (x) = (x + 2) (x - 1) (3 - x)

then the x-intercepts come from setting each factor equal to zero.

Cubic graph shape

A cubic is a polynomial with highest power $x^3$ . Its end behaviour depends on the sign of the $x^3$ coefficient:

positive $x^3$ coefficient: down on the left, up on the right
negative $x^3$ coefficient: up on the left, down on the right

If a factor is repeated, for example $x-2)^2$ , the graph usually touches the x-axis at that root rather than crossing it.

Example

Sketching a factorised cubic

Sketch $y = (x + 2) (x - 1) (3 - x)$ , showing where it meets the coordinate axes.

Find the x-intercepts by setting each factor equal to zero.
$x+2=0⇒x=−2x−1=0⇒x=13−x=0⇒x=3\begin{aligned} x+2&=0 \Rightarrow x=-2\\ x-1&=0 \Rightarrow x=1\\ 3-x&=0 \Rightarrow x=3 \end{aligned}$
Write the x-intercepts as coordinates.
$(−2,0),(1,0),(3,0)(-2,0),\quad (1,0),\quad (3,0)$
Find the y-intercept by substituting $x = 0$ .
$\times (-1)\times 3=-6$
Decide the overall cubic shape. The leading term is negative, because $\times x \times (-x)=-x^3$ , so the curve goes up on the left and down on the right.
Draw a smooth cubic crossing the x-axis at $x = - 2$ , $x = 1$ and $x = 3$ , and crossing the y-axis at $(0, - 6)$ .

A sketch of the cubic showing its three x-intercepts, y-intercept and negative leading-coefficient end behaviour.

Tip

Sketch before you perfect

In sketch questions, the shape and labelled intercepts are usually worth more than scale accuracy. Your graph does not need squared-paper precision unless the question asks for it.

2. Function notation and transformations

Definition

Function notation

Writing $y = f (x)$ means that the y-value is found by putting $x$ into a rule called $f$ . For example, if $f(x)=x^2+3$ , then $f (2) = 7$ .

A transformation changes the position or shape of a graph. In this topic, the main transformations are translations, reflections and stretches.

A translation slides every point by the same amount.
A reflection flips a graph in a mirror line, usually an axis.
A stretch multiplies distances from an axis by a scale factor.

The diagram below shows the big idea: outside the function affects the y-values; inside the function affects the x-values.

A summary diagram comparing vertical and horizontal transformations of the same base graph.

Four common transformations of a cubic graph: vertical translation, horizontal translation, reflection and horizontal compression

Key transformation rules

For a point $(x, y)$ on $y = f (x)$ :

$y = f (x) + a$ moves the graph up by $a$ .
$y = f (x) - a$ moves the graph down by $a$ .
$y = f (x - a)$ moves the graph right by $a$ .
$y = f (x + a)$ moves the graph left by $a$ .
$y = - f (x)$ reflects the graph in the x-axis.
$y = f (- x)$ reflects the graph in the y-axis.
$y = a f (x)$ is a stretch parallel to the y-axis, scale factor $a$ .
$y = f (a x)$ is a stretch parallel to the x-axis, scale factor $1a\frac{1}{a}$ .

Key Idea

Outside vs inside

Changes outside the bracket affect y-values in the obvious way. Changes inside the bracket affect x-values in the opposite way.

Common Mistake

Moving the wrong way

The graph of $y = f (x - 3)$ moves right by 3, not left. Think: to get the same output as $f (0)$ , you need $x - 3 = 0$ , so $x = 3$ .

Example

Transforming labelled points

A curve $y = f (x)$ passes through $(- 2, 0)$ , $(1, 0)$ , $(4, 0)$ and $(0, 8)$ . Find the corresponding points on $y = f (x - 3)$ and $y = f (- x)$ .

The labelled points on a graph shift right for f(x-3) and reflect across the y-axis for f(-x).

For $y = f (x - 3)$ , the graph moves 3 units to the right. Add 3 to every x-coordinate.
$(−2,0)→(1,0),(1,0)→(4,0),(4,0)→(7,0),(0,8)→(3,8)(-2,0)\to(1,0),\quad (1,0)\to(4,0),\quad (4,0)\to(7,0),\quad (0,8)\to(3,8)$
For $y = f (- x)$ , the graph reflects in the y-axis. Change the sign of every x-coordinate.
$(−2,0)→(2,0),(1,0)→(−1,0),(4,0)→(−4,0),(0,8)→(0,8)(-2,0)\to(2,0),\quad (1,0)\to(-1,0),\quad (4,0)\to(-4,0),\quad (0,8)\to(0,8)$
Notice that the y-intercept stays on the y-axis under $y = f (- x)$ , because its x-coordinate is zero.

3. Quadratics: completing the square and minimum points

A quadratic has highest power $x^2$ . If the coefficient of $x^2$ is positive, it has a minimum point.

Completing the square rewrites a quadratic in the form

x+a)^2+b

This makes the turning point easier to read.

Example

Minimum point and transformed minimum points

Let $f(x)=x^2+6x+11$ .

Complete the square.
$f(x)=x2+6x+11=(x+3)2−9+11=(x+3)2+2\begin{aligned} f(x)&=x^2+6x+11\\ &=(x+3)^2-9+11\\ &=(x+3)^2+2 \end{aligned}$
Read off the minimum point. Since $x+3)^2$ is smallest when $x = - 3$ , the minimum point is $(- 3, 2)$ .

Completing the square reveals the vertex of the quadratic as a translation of y=x^2.

For $y = 2 f (x)$ , all y-values are doubled. The minimum point becomes $(- 3, 4)$ .
For $y = f (2 x)$ , the graph is stretched parallel to the x-axis with scale factor $12\frac{1}{2}$ . Halve the x-coordinate, so the minimum point becomes $(−32,2)\left(-\frac{3}{2},2\right)$ .

Common Mistake

Stretches do not always move both coordinates

For $y = 2 f (x)$ , only y-values change. For $y = f (2 x)$ , only x-values change. Do not double or halve both coordinates.

4. Reciprocal graphs and asymptotes

The basic reciprocal graph $y=1xy=\frac{1}{x}$ has two branches. It has:

vertical asymptote $x = 0$
horizontal asymptote $y = 0$

Translations move the asymptotes as well as the curve.

For example, $y=1x−2+1y=\frac{1}{x-2}+1$ is $y=1xy=\frac{1}{x}$ translated 2 units right and 1 unit up.

Reciprocal graph y equals 1 over x minus 2 plus 1, with asymptotes x equals 2 and y equals 1

Example

Sketching a translated reciprocal graph

Sketch $y=2x−1+3y=\frac{2}{x-1}+3$ , giving the intercepts and asymptotes.

Find the vertical asymptote from the denominator.
$\Rightarrow x=1$
Find the horizontal asymptote from the vertical shift. Since 3 is added outside the fraction, the horizontal asymptote is $y = 3$ .
Find the y-intercept by substituting $x = 0$ .
$y=20−1+3=−2+3=1y=\frac{2}{0-1}+3=-2+3=1$
Find the x-intercept by setting $y = 0$ .
$0=2x−1+3−3=2x−1−3(x−1)=2−3x+3=2x=13\begin{aligned} 0&=\frac{2}{x-1}+3\\ -3&=\frac{2}{x-1}\\ -3(x-1)&=2\\ -3x+3&=2\\ x&=\frac{1}{3} \end{aligned}$
Label the intercepts $(0, 1)$ and $(13,0)\left(\frac{1}{3},0\right)$ , then draw two branches approaching $x = 1$ and $y = 3$ .

A translated reciprocal graph has asymptotes shifted to x=1 and y=3, with intercepts labelled.

5. Using sketches for equations and inequalities

A graph can help you solve equations and inequalities.

Solving $f (x) = 0$ means finding where the graph meets the x-axis.
Solving $f (x) > 0$ means finding where the graph is above the x-axis.
Solving $f(x)≤0f(x)\leq 0$ means finding where the graph is on or below the x-axis.

Example

Solving an inequality from a cubic sketch

Let $f(x)=(x+1)(x-2)^2$ . Use the graph to solve $f(x)≤0f(x)\leq 0$ .

Find the roots.
$x=−1,x=2x=-1,\quad x=2$
Notice that $x = 2$ is a repeated root because of $x-2)^2$ , so the graph touches the x-axis there.
Work out the overall shape. The leading term is positive, so the cubic goes down on the left and up on the right.
Decide where the graph is on or below the x-axis. It is below the axis to the left of $x = - 1$ , crosses at $x = - 1$ , then stays above the axis except for touching at $x = 2$ .

The cubic crosses at the simple root x=-1 but only touches the axis at the repeated root x=2.

Write the solution in set notation.
${x:x≤−1}∪{2}\{x:x\leq -1\}\cup\{2\}$

Tip

Repeated roots and inequalities

At an even repeated root, the graph touches the x-axis and turns around. The sign of $f (x)$ usually does not change there.

6. Equations from intersections

If two graphs intersect, their y-values are equal at the intersection points. So the number of real solutions to an equation can often be read from the number of intersections.

Example

Counting real solutions by sketching

Sketch ideas are enough to state how many real solutions there are to

x2+4x+1x=0x^2+4x+\frac{1}{x}=0

Rearrange the equation as an intersection problem.
$x2+4x=−1xx^2+4x=-\frac{1}{x}$
Think of sketching $y=x^2+4x$ and $y=−1xy=-\frac{1}{x}$ on the same axes.

Sketching the quadratic and reciprocal curve on the same axes lets you count the real solutions from their intersections.

The quadratic $y=x^2+4x$ has roots at $x = 0$ and $x = - 4$ , and its minimum is below the x-axis.
The reciprocal curve $y=−1xy=-\frac{1}{x}$ has one branch in quadrant 2 and one branch in quadrant 4, with asymptotes $x = 0$ and $y = 0$ .
Count the intersections from the sketch. Each intersection gives one real solution to the original equation.

Exam technique

In the exam

Start every sketch by finding the x-intercepts and y-intercept; label them clearly.
For transformations, track key points such as intercepts and turning points rather than trying to redraw from scratch.
Remember: $f (x - a)$ moves right by $a$ , while $f (a x)$ changes x-coordinates by scale factor $1a\frac{1}{a}$ .

Self review

Check yourself

If $f (x)$ has a minimum at $(4, 2)$ , where is the minimum of $y = f (2 x)$ ?
What are the asymptotes of $y=3x+5−1y=\frac{3}{x+5}-1$ ?
Why does a graph usually touch, rather than cross, the x-axis at a repeated root such as $x-3)^2$ ?

Graphs and Transformations

What you'll learn

How to sketch polynomial and reciprocal graphs from their equations.
How to find and label intercepts, roots, turning points and asymptotes.
How transformations such as $f (x) + a$ , $f (x - a)$ , $- f (x)$ and $f (a x)$ change a graph.
How to use a sketch to solve equations and inequalities.

1. Reading a graph: the essential vocabulary

A graph is a picture of all the points $(x, y)$ that satisfy an equation, such as $y = f (x)$ .

Definition

Intercepts, roots and asymptotes

An x-intercept is where a graph crosses or touches the x-axis, so $y = 0$ .
A y-intercept is where a graph crosses the y-axis, so $x = 0$ .
A root or zero of $f (x)$ is a value of $x$ where $f (x) = 0$ .
An asymptote is a straight line that a curve gets closer and closer to but does not meet.

For factorised polynomial graphs, the roots are usually the easiest part. If

f (x) = (x + 2) (x - 1) (3 - x)

then the x-intercepts come from setting each factor equal to zero.

Cubic graph shape

A cubic is a polynomial with highest power $x^3$ . Its end behaviour depends on the sign of the $x^3$ coefficient:

positive $x^3$ coefficient: down on the left, up on the right
negative $x^3$ coefficient: up on the left, down on the right

If a factor is repeated, for example $x-2)^2$ , the graph usually touches the x-axis at that root rather than crossing it.

Example

Sketching a factorised cubic

Sketch $y = (x + 2) (x - 1) (3 - x)$ , showing where it meets the coordinate axes.

Find the x-intercepts by setting each factor equal to zero.
$x+2=0⇒x=−2x−1=0⇒x=13−x=0⇒x=3\begin{aligned} x+2&=0 \Rightarrow x=-2\\ x-1&=0 \Rightarrow x=1\\ 3-x&=0 \Rightarrow x=3 \end{aligned}$
Write the x-intercepts as coordinates.
$(−2,0),(1,0),(3,0)(-2,0),\quad (1,0),\quad (3,0)$
Find the y-intercept by substituting $x = 0$ .
$\times (-1)\times 3=-6$
Decide the overall cubic shape. The leading term is negative, because $\times x \times (-x)=-x^3$ , so the curve goes up on the left and down on the right.
Draw a smooth cubic crossing the x-axis at $x = - 2$ , $x = 1$ and $x = 3$ , and crossing the y-axis at $(0, - 6)$ .

A sketch of the cubic showing its three x-intercepts, y-intercept and negative leading-coefficient end behaviour.

Tip

Sketch before you perfect

In sketch questions, the shape and labelled intercepts are usually worth more than scale accuracy. Your graph does not need squared-paper precision unless the question asks for it.

2. Function notation and transformations

Definition

Function notation

Writing $y = f (x)$ means that the y-value is found by putting $x$ into a rule called $f$ . For example, if $f(x)=x^2+3$ , then $f (2) = 7$ .

A transformation changes the position or shape of a graph. In this topic, the main transformations are translations, reflections and stretches.

A translation slides every point by the same amount.
A reflection flips a graph in a mirror line, usually an axis.
A stretch multiplies distances from an axis by a scale factor.

The diagram below shows the big idea: outside the function affects the y-values; inside the function affects the x-values.

A summary diagram comparing vertical and horizontal transformations of the same base graph.

Four common transformations of a cubic graph: vertical translation, horizontal translation, reflection and horizontal compression

Key transformation rules

For a point $(x, y)$ on $y = f (x)$ :

$y = f (x) + a$ moves the graph up by $a$ .
$y = f (x) - a$ moves the graph down by $a$ .
$y = f (x - a)$ moves the graph right by $a$ .
$y = f (x + a)$ moves the graph left by $a$ .
$y = - f (x)$ reflects the graph in the x-axis.
$y = f (- x)$ reflects the graph in the y-axis.
$y = a f (x)$ is a stretch parallel to the y-axis, scale factor $a$ .
$y = f (a x)$ is a stretch parallel to the x-axis, scale factor $1a\frac{1}{a}$ .

Key Idea

Outside vs inside

Changes outside the bracket affect y-values in the obvious way. Changes inside the bracket affect x-values in the opposite way.

Common Mistake

Moving the wrong way

The graph of $y = f (x - 3)$ moves right by 3, not left. Think: to get the same output as $f (0)$ , you need $x - 3 = 0$ , so $x = 3$ .

Example

Transforming labelled points

A curve $y = f (x)$ passes through $(- 2, 0)$ , $(1, 0)$ , $(4, 0)$ and $(0, 8)$ . Find the corresponding points on $y = f (x - 3)$ and $y = f (- x)$ .

The labelled points on a graph shift right for f(x-3) and reflect across the y-axis for f(-x).

For $y = f (x - 3)$ , the graph moves 3 units to the right. Add 3 to every x-coordinate.
$(−2,0)→(1,0),(1,0)→(4,0),(4,0)→(7,0),(0,8)→(3,8)(-2,0)\to(1,0),\quad (1,0)\to(4,0),\quad (4,0)\to(7,0),\quad (0,8)\to(3,8)$
For $y = f (- x)$ , the graph reflects in the y-axis. Change the sign of every x-coordinate.
$(−2,0)→(2,0),(1,0)→(−1,0),(4,0)→(−4,0),(0,8)→(0,8)(-2,0)\to(2,0),\quad (1,0)\to(-1,0),\quad (4,0)\to(-4,0),\quad (0,8)\to(0,8)$
Notice that the y-intercept stays on the y-axis under $y = f (- x)$ , because its x-coordinate is zero.

3. Quadratics: completing the square and minimum points

A quadratic has highest power $x^2$ . If the coefficient of $x^2$ is positive, it has a minimum point.

Completing the square rewrites a quadratic in the form

x+a)^2+b

This makes the turning point easier to read.

Example

Minimum point and transformed minimum points

Let $f(x)=x^2+6x+11$ .

Complete the square.
$f(x)=x2+6x+11=(x+3)2−9+11=(x+3)2+2\begin{aligned} f(x)&=x^2+6x+11\\ &=(x+3)^2-9+11\\ &=(x+3)^2+2 \end{aligned}$
Read off the minimum point. Since $x+3)^2$ is smallest when $x = - 3$ , the minimum point is $(- 3, 2)$ .

Completing the square reveals the vertex of the quadratic as a translation of y=x^2.

For $y = 2 f (x)$ , all y-values are doubled. The minimum point becomes $(- 3, 4)$ .
For $y = f (2 x)$ , the graph is stretched parallel to the x-axis with scale factor $12\frac{1}{2}$ . Halve the x-coordinate, so the minimum point becomes $(−32,2)\left(-\frac{3}{2},2\right)$ .

Common Mistake

Stretches do not always move both coordinates

For $y = 2 f (x)$ , only y-values change. For $y = f (2 x)$ , only x-values change. Do not double or halve both coordinates.

4. Reciprocal graphs and asymptotes

The basic reciprocal graph $y=1xy=\frac{1}{x}$ has two branches. It has:

vertical asymptote $x = 0$
horizontal asymptote $y = 0$

Translations move the asymptotes as well as the curve.

For example, $y=1x−2+1y=\frac{1}{x-2}+1$ is $y=1xy=\frac{1}{x}$ translated 2 units right and 1 unit up.

Reciprocal graph y equals 1 over x minus 2 plus 1, with asymptotes x equals 2 and y equals 1

Example

Sketching a translated reciprocal graph

Sketch $y=2x−1+3y=\frac{2}{x-1}+3$ , giving the intercepts and asymptotes.

Find the vertical asymptote from the denominator.
$\Rightarrow x=1$
Find the horizontal asymptote from the vertical shift. Since 3 is added outside the fraction, the horizontal asymptote is $y = 3$ .
Find the y-intercept by substituting $x = 0$ .
$y=20−1+3=−2+3=1y=\frac{2}{0-1}+3=-2+3=1$
Find the x-intercept by setting $y = 0$ .
$0=2x−1+3−3=2x−1−3(x−1)=2−3x+3=2x=13\begin{aligned} 0&=\frac{2}{x-1}+3\\ -3&=\frac{2}{x-1}\\ -3(x-1)&=2\\ -3x+3&=2\\ x&=\frac{1}{3} \end{aligned}$
Label the intercepts $(0, 1)$ and $(13,0)\left(\frac{1}{3},0\right)$ , then draw two branches approaching $x = 1$ and $y = 3$ .

A translated reciprocal graph has asymptotes shifted to x=1 and y=3, with intercepts labelled.

5. Using sketches for equations and inequalities

A graph can help you solve equations and inequalities.

Solving $f (x) = 0$ means finding where the graph meets the x-axis.
Solving $f (x) > 0$ means finding where the graph is above the x-axis.
Solving $f(x)≤0f(x)\leq 0$ means finding where the graph is on or below the x-axis.

Example

Solving an inequality from a cubic sketch

Let $f(x)=(x+1)(x-2)^2$ . Use the graph to solve $f(x)≤0f(x)\leq 0$ .

Find the roots.
$x=−1,x=2x=-1,\quad x=2$
Notice that $x = 2$ is a repeated root because of $x-2)^2$ , so the graph touches the x-axis there.
Work out the overall shape. The leading term is positive, so the cubic goes down on the left and up on the right.
Decide where the graph is on or below the x-axis. It is below the axis to the left of $x = - 1$ , crosses at $x = - 1$ , then stays above the axis except for touching at $x = 2$ .

The cubic crosses at the simple root x=-1 but only touches the axis at the repeated root x=2.

Write the solution in set notation.
${x:x≤−1}∪{2}\{x:x\leq -1\}\cup\{2\}$

Tip

Repeated roots and inequalities

At an even repeated root, the graph touches the x-axis and turns around. The sign of $f (x)$ usually does not change there.

6. Equations from intersections

If two graphs intersect, their y-values are equal at the intersection points. So the number of real solutions to an equation can often be read from the number of intersections.

Example

Counting real solutions by sketching

Sketch ideas are enough to state how many real solutions there are to

x2+4x+1x=0x^2+4x+\frac{1}{x}=0

Rearrange the equation as an intersection problem.
$x2+4x=−1xx^2+4x=-\frac{1}{x}$
Think of sketching $y=x^2+4x$ and $y=−1xy=-\frac{1}{x}$ on the same axes.

Sketching the quadratic and reciprocal curve on the same axes lets you count the real solutions from their intersections.

The quadratic $y=x^2+4x$ has roots at $x = 0$ and $x = - 4$ , and its minimum is below the x-axis.
The reciprocal curve $y=−1xy=-\frac{1}{x}$ has one branch in quadrant 2 and one branch in quadrant 4, with asymptotes $x = 0$ and $y = 0$ .
Count the intersections from the sketch. Each intersection gives one real solution to the original equation.

Exam technique

In the exam

Start every sketch by finding the x-intercepts and y-intercept; label them clearly.
For transformations, track key points such as intercepts and turning points rather than trying to redraw from scratch.
Remember: $f (x - a)$ moves right by $a$ , while $f (a x)$ changes x-coordinates by scale factor $1a\frac{1}{a}$ .

Self review

Check yourself

If $f (x)$ has a minimum at $(4, 2)$ , where is the minimum of $y = f (2 x)$ ?
What are the asymptotes of $y=3x+5−1y=\frac{3}{x+5}-1$ ?
Why does a graph usually touch, rather than cross, the x-axis at a repeated root such as $x-3)^2$ ?

Graphs and Transformations

What you'll learn

1. Reading a graph: the essential vocabulary

Cubic graph shape

2. Function notation and transformations

Key transformation rules

3. Quadratics: completing the square and minimum points

4. Reciprocal graphs and asymptotes

5. Using sketches for equations and inequalities

6. Equations from intersections

Recap questions

You've reached the end

Graphs and Transformations

What you'll learn

1. Reading a graph: the essential vocabulary

Cubic graph shape

2. Function notation and transformations

Key transformation rules

3. Quadratics: completing the square and minimum points

4. Reciprocal graphs and asymptotes

5. Using sketches for equations and inequalities

6. Equations from intersections

Recap questions

You've reached the end

Graphs and Transformations

What you'll learn

1. Reading a graph: the essential vocabulary

Cubic graph shape

2. Function notation and transformations

Key transformation rules

3. Quadratics: completing the square and minimum points

4. Reciprocal graphs and asymptotes

5. Using sketches for equations and inequalities

6. Equations from intersections

Recap questions

You've reached the end

Graphs and Transformations Revision Guide

Graphs and Transformations

What you'll learn

1. Reading a graph: the essential vocabulary

Cubic graph shape

2. Function notation and transformations

Key transformation rules

3. Quadratics: completing the square and minimum points

4. Reciprocal graphs and asymptotes

5. Using sketches for equations and inequalities

6. Equations from intersections

Recap questions

You've reached the end

Graphs and Transformations Revision Guide