What transformation changes \(y = x^2\) to \(y = x^2 + 3\)?
Adding outside the bracket moves the graph vertically.
\(y = x^2 + 3\) shifts the graph up by 3 units.
Graph Transformations
Graph transformations change the position or shape of a graph through translations, reflections and stretches. They are closely linked to straight line graphs and quadratic graphs, and are key to understanding function behaviour.
Overview
Graph transformations show how a graph changes when its equation changes.
A graph can move left, right, up or down, reflect in an axis, or be stretched and compressed.
Original: \( y = f(x) \)
Transformed: \( y = f(x - 3) + 2 \)
Transformed: \( y = f(x - 3) + 2 \)
This means the graph has moved 3 units to the right and 2 units up.
What you should understand after this topic
- Understand how translations change the position of a graph
- Understand how reflections transform a graph
- Understand how stretches change the shape of a graph
- Recognise the difference between changes inside and outside brackets
- Describe a transformed graph clearly
Key Definitions
Translation
A movement of the whole graph without changing its shape.
Reflection
A flip of the graph in an axis.
Stretch
A transformation that changes the size of the graph in one direction.
Scale Factor
The number that tells you how much the graph is stretched.
Invariant Point
A point that stays in the same place after a transformation.
Inside vs Outside
Changes inside brackets affect horizontal movement, while changes outside affect vertical movement.
Key Rules
\( y = f(x) + a \)
Move the graph <strong>up</strong> by \( a \).
\( y = f(x) - a \)
Move the graph <strong>down</strong> by \( a \).
\( y = f(x - a) \)
Move the graph <strong>right</strong> by \( a \).
\( y = f(x + a) \)
Move the graph <strong>left</strong> by \( a \).
\( y = -f(x) \)
Reflect in the <strong>\(x\)-axis</strong>.
\( y = f(-x) \)
Reflect in the <strong>\(y\)-axis</strong>.
\( y = af(x) \)
Stretch parallel to the \(y\)-axis by scale factor \( a \).
\( y = f(ax) \)
Stretch parallel to the \(x\)-axis by scale factor \( \frac{1}{a} \).
Golden Rule
Outside the brackets works the normal way. Inside the brackets works the opposite way.
How to Solve
Step 1: Start with the base graph
All transformation questions begin with a known graph.
\( y = x^2 \), \( y = x^3 \), \( y = |x| \), \( y = \sin x \)
The new equation shows how this original graph has changed.
Exam tip: Always identify the original graph first before describing any transformation.
Step 2: Vertical translations (up and down)
Changes outside the function move the graph vertically.
\( y = f(x) + 4 \)
\( y = f(x) - 2 \)
Add → move up
Subtract → move down
Exam thinking: Look at the number outside the brackets.
Step 3: Horizontal translations (left and right)
Changes inside the function move the graph horizontally.
\( y = f(x - 5) \)
\( y = f(x + 2) \)
Subtract → move right
Add → move left
Key rule
Inside the brackets is reversed:
\( x - 3 \) → right 3
\( x + 3 \) → left 3
Step 4: Reflections
Reflections flip the graph.
\( y = -f(x) \)
\( y = f(-x) \)
Negative outside → reflection in the x-axis
Negative inside → reflection in the y-axis
Exam tip: Check whether the negative is inside or outside.
Step 5: Stretches
Multiplying changes the size of the graph.
\( y = 3f(x) \)
\( y = f(2x) \)
Outside → vertical stretch (away from x-axis)
Inside → horizontal stretch (scale factor is reversed)
- If the number is outside, multiply all y-values.
- If the number is inside, divide x-values by that number.
- For \( f(2x) \), the scale factor is \( \frac{1}{2} \).
Step 6: Combining transformations
Many exam questions combine multiple transformations.
\( y = 2f(x - 3) + 1 \)
Exam tip: Always apply transformations in this order: inside → multiply → add.
- Start with the inside: \( x - 3 \) → right 3
- Then multiply: \( 2f(x) \) → vertical stretch
- Finally add: \( +1 \) → up 1
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on identifying and describing transformations of graphs.
Edexcel
Describe the transformation that maps the graph of \( y = x^2 \) onto the graph of \( y = x^2 + 3 \).
Edexcel
Describe the transformation that maps the graph of \( y = x^2 \) onto the graph of \( y = (x - 4)^2 \).
Edexcel
Describe the transformation that maps the graph of \( y = x^2 \) onto the graph of \( y = (x + 2)^2 \).
Edexcel
Write down the coordinates of the turning point of \( y = (x - 3)^2 + 1 \).
Edexcel
The graph of \( y = f(x) \) is translated by the vector \( \begin{pmatrix} 2 \\ -3 \end{pmatrix} \). Write down the equation of the new graph.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on reflections, stretches, and combined transformations.
AQA
Describe the transformation that maps \( y = f(x) \) to \( y = -f(x) \).
AQA
Describe the transformation that maps \( y = f(x) \) to \( y = f(-x) \).
AQA
Describe the transformation that maps \( y = f(x) \) to \( y = 3f(x) \).
AQA
Describe the transformation that maps \( y = f(x) \) to \( y = f(2x) \).
AQA
The graph of \( y = x^2 \) is transformed to \( y = -2(x - 1)^2 + 4 \). Describe the sequence of transformations.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, interpretation, and sketching transformed graphs.
OCR
Sketch the graph of \( y = x^2 \) and the graph of \( y = x^2 - 4 \) on the same axes.
OCR
Sketch the graph of \( y = x^2 \) and \( y = (x + 1)^2 \).
OCR
The graph of \( y = f(x) \) passes through the point (2, 5). Find the corresponding point on the graph of \( y = f(x) + 3 \).
OCR
The graph of \( y = f(x) \) passes through the point (-1, 4). Find the corresponding point on the graph of \( y = f(x - 2) \).
OCR
Explain how the graph of y = f(x) changes when it is transformed to y = -f(x) + 2.
Exam Checklist
Step 1
Check whether the change is inside or outside the brackets.
Step 2
Decide whether it is a translation, reflection or stretch.
Step 3
Be careful with direction for horizontal changes.
Step 4
Use the correct exam wording, especially for stretches.
Most common exam mistakes
Inside brackets
Writing left instead of right, or right instead of left.
Reflections
Mixing up \( y=-f(x) \) and \( y=f(-x) \).
Stretches
Forgetting the phrase “parallel to the axis”.
Horizontal stretches
Using the wrong scale factor.
Common Mistakes
These are common mistakes students make when transforming graphs in GCSE Maths.
Getting horizontal direction wrong
Incorrect
A student says \(y = f(x - 3)\) shifts the graph left.
Correct
Inside changes work in the opposite direction. \(y = f(x - 3)\) shifts the graph 3 units to the right.
Mixing up reflections in axes
Incorrect
A student confuses reflection in the x-axis with reflection in the y-axis.
Correct
Reflection in the x-axis changes \(y = f(x)\) to \(y = -f(x)\). Reflection in the y-axis changes \(y = f(x)\) to \(y = f(-x)\).
Confusing inside and outside changes
Incorrect
A student treats all transformations the same.
Correct
Changes outside the function (e.g. \(y = f(x) + 3\)) affect the graph vertically. Changes inside (e.g. \(y = f(x - 3)\)) affect it horizontally.
Misunderstanding horizontal transformations
Incorrect
A student assumes inside changes behave like normal scaling.
Correct
Horizontal transformations work in the opposite way. For example, \(y = f(2x)\) compresses the graph horizontally, not stretches it.
Using the wrong scale factor for stretches
Incorrect
A student applies the scale factor incorrectly for horizontal stretches.
Correct
For horizontal stretches, the scale factor is the reciprocal of the number inside the function. For example, \(y = f(0.5x)\) stretches the graph horizontally by a factor of 2.
Try It Yourself
Practise transforming graphs using translations, reflections and stretches.
Foundation
Foundation Practice
Understand translations and simple reflections.
Describe the transformation from \(y = x^2\) to \(y = x^2 - 4\).
Subtracting moves the graph down.
The graph is translated down 4 units.
What transformation changes \(y = x^2\) to \(y = (x - 2)^2\)?
Inside the bracket works opposite.
\(x - 2\) moves the graph 2 units to the right.
Describe the transformation from \(y = x^2\) to \(y = (x + 3)^2\).
Inside the bracket reverses direction.
The graph is translated 3 units to the left.
What does \(y = -x^2\) represent?
A negative in front flips the graph.
The graph is reflected in the x-axis.
Describe the transformation from \(y = x^2\) to \(y = -x^2\).
Look at the negative sign.
The graph is reflected in the x-axis.
A student says \(y = (x - 2)^2\) moves left 2. What is wrong?
Inside brackets is reversed.
\(x - 2\) moves the graph right, not left.
What transformation changes \(y = x^2\) to \(y = x^2 + 5\)?
Outside the bracket shifts vertically.
The graph moves up 5 units.
Which transformation moves a graph down?
Look for subtraction outside.
\(y = x^2 - 2\) moves the graph down 2 units.
Describe the transformation from \(y = x^2\) to \(y = x^2 + 1\).
Look at the constant.
The graph is translated up 1 unit.
Higher
Higher Practice
Understand stretches, combined transformations and function notation.
What transformation changes \(y = x^2\) to \(y = 3x^2\)?
Multiplying outside stretches vertically.
Multiplying by 3 stretches the graph vertically by scale factor 3.
Describe the transformation from \(y = x^2\) to \(y = 2x^2\).
Look at the number multiplying the function.
The graph is stretched vertically by scale factor 2.
What transformation changes \(y = x^2\) to \(y = x^2 - 3\)?
Subtracting moves down.
The graph moves down 3 units.
Describe the transformation from \(y = x^2\) to \(y = (x - 4)^2 + 1\).
Inside = horizontal, outside = vertical.
Right 4 units and up 1 unit.
What transformation changes \(y = x^2\) to \(y = -(x - 1)^2\)?
Combine reflection and translation.
The graph is reflected in the x-axis and shifted right 1.
Describe the transformation from \(y = x^2\) to \(y = (x + 2)^2 - 3\).
Inside and outside shifts.
Left 2 and down 3.
A student says \(y = 2(x - 1)^2\) is a horizontal stretch. What is wrong?
Multiplication outside affects y-values.
Multiplying outside causes a vertical stretch.
Describe the transformation from \(y = x^2\) to \(y = -2x^2\).
Look at both sign and number.
Reflection in x-axis and vertical stretch by factor 2.
Which transformation moves a graph left?
Inside bracket reverses direction.
\(x + 3\) moves the graph left 3 units.
Describe the transformation from \(y = x^2\) to \(y = 3(x + 1)^2 - 2\).
Break into parts.
Left 1, vertical stretch by 3, then down 2.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is a graph transformation?
A change in the position or shape of a graph.
What does f(x)+a do?
It shifts the graph vertically.
What does f(x+a) do?
It shifts the graph horizontally in the opposite direction.