What are the different types of transformations in mathematics?

In mathematics, particularly in the context of IB MYP Algebra, transformations refer to operations that alter the position or size of a shape or graph. The main types of transformations include translations (sliding), rotations (turning), reflections (flipping), and dilations (resizing). Each transformation has specific rules that determine how the shape or graph is altered.

How do you perform a translation on a coordinate plane?

To perform a translation on a coordinate plane, you move every point of a shape or graph a certain distance in a specified direction. This is typically described using a vector, such as (a, b), where 'a' represents the horizontal shift and 'b' represents the vertical shift. For example, translating a point (x, y) by the vector (3, -2) results in the new point (x+3, y-2).

What is the effect of a reflection transformation on a graph?

A reflection transformation flips a graph or shape over a specified line, known as the line of reflection. Common lines of reflection include the x-axis, y-axis, or y=x line. For example, reflecting a point (x, y) over the x-axis results in the point (x, -y). This transformation changes the orientation of the shape but not its size or shape.

How does a dilation transformation affect a shape?

A dilation transformation changes the size of a shape while maintaining its proportions. It involves a scale factor, which determines how much the shape is enlarged or reduced. If the scale factor is greater than 1, the shape enlarges; if it is between 0 and 1, the shape reduces. The center of dilation is the point from which the shape is expanded or contracted.

Can you explain how rotation transformations work in algebra?

Rotation transformations involve turning a shape around a fixed point, known as the center of rotation, by a certain angle and direction (clockwise or counterclockwise). In the IB MYP curriculum, rotations are often performed around the origin (0,0). For example, rotating a point (x, y) 90 degrees counterclockwise around the origin results in the new point (-y, x). The shape's size and proportions remain unchanged during rotation.

Why is "Saying $f(x - 3)$ shifts the graph LEFT." a common mistake in Transformation?

Why it happens: The minus 'feels' like negative direction. How to avoid it: f(x - 3) shifts the graph RIGHT by 3. Think of it as 'the input is x - 3, which equals 0 when x = 3' — so the original f(0) now appears at x = 3.

Why is "Saying $f(2x)$ stretches horizontally by 2." a common mistake in Transformation?

Why it happens: Multiplying suggests enlargement. How to avoid it: f(2x) COMPRESSES horizontally by factor 2 (i.e. half as wide). For horizontal stretch by 2, use f(x/2).

AlgebraIB MYP Maths Extended Level

Transformation

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Transformation (of Graphs) — IB MYP Mathematics (Extended): translation, reflection, stretch

We can shift, flip and stretch graphs by modifying the function. This MYP Mathematics Extended note covers the four fundamental graph transformations.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Apply translations, reflections and stretches to a graph.
MYP Mathematics A — Identify the transformation from the equation.
MYP Mathematics C — Sketch the transformed graph.

Translations

Shift the graph without changing its shape.

Vertical translation: $y = f(x) + c$ .

$c > 0$ : shift UP by $c$ .
$c < 0$ : shift DOWN by $|c|$ .

Horizontal translation: $y = f(x - c)$ .

$c > 0$ : shift RIGHT by $c$ .
$c < 0$ : shift LEFT by $|c|$ .

Common trap: $f(x - c)$ shifts RIGHT (positive direction) even though the algebra LOOKS negative. Think of it as 'whatever was happening at $x = 0$ now happens at $x = c$ '.

Worked example. Sketch $y = x^2 - 4$ given the graph of $y = x^2$ .

Subtracting 4 from the output → shift DOWN by 4.
Vertex of $y = x^2$ is $(0, 0)$ . After shift: vertex at $(0, -4)$ .

Worked example. Sketch $y = (x - 3)^2$ given $y = x^2$ .

$(x - 3)^2$ has $x - 3$ in place of $x$ → shift RIGHT by 3.
Vertex moves from $(0, 0)$ to $(3, 0)$ .

Combined. $y = (x - 3)^2 - 4$ : shift RIGHT 3 then DOWN 4. Vertex at $(3, -4)$ .

This is why $y = a(x - h)^2 + k$ is called vertex form — the vertex is at $(h, k)$ .

$f(x) + c$ : vertical shift.
$f(x - c)$ : horizontal shift RIGHT (even with the minus).
Vertex form: $y = a(x - h)^2 + k$ has vertex $(h, k)$ .

Reflections and stretches

Flip and scale.

Reflection in x-axis: $y = -f(x)$ . Every $y$ becomes $-y$ .

Reflection in y-axis: $y = f(-x)$ . Every $x$ becomes $-x$ .

Vertical stretch by factor $a$ : $y = a \cdot f(x)$ . Y-values scale by $a$ .

$a > 1$ : stretches tall.
$0 < a < 1$ : squashes flat.
$a < 0$ : also reflects in x-axis.

Horizontal stretch by factor $1/a$ : $y = f(ax)$ . X-values scale by $1/a$ .

$a > 1$ : squashes narrow.
$0 < a < 1$ : stretches wide.

(Notice: horizontal stretches are 'backwards' — multiplying $x$ by 2 SQUEEZES the graph by half, not doubles it.)

Translations preserve shape. Adding/subtracting a constant outside the function shifts vertically; subtracting from $x$ inside the bracket shifts right.

Combining transformations. Apply them in order:

Inside the function (horizontal effects): scale, reflect, translate.
Outside (vertical effects): scale, reflect, translate.

Worked example. Describe $y = -2(x + 1)^2 + 3$ as a transformation of $y = x^2$ .

$(x + 1)$ : shift LEFT 1.
$-2 \cdot$ : vertical stretch by 2, then reflection in x-axis.
$+ 3$ : shift UP 3.
Vertex: starts at $(0, 0)$ , shifts to $(-1, 3)$ , opens DOWNWARD with stretched shape.

$-f(x)$ : reflect in x-axis.
$f(-x)$ : reflect in y-axis.
$af(x)$ : vertical stretch (factor $a$ ).
$f(ax)$ : horizontal stretch (factor $1/a$ ).

How it’s examined

Criterion A: identify transformation and apply. Criterion C: sketch transformed graph clearly.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Transformation

Step-by-step solutions to past-paper-style questions on transformation, written exactly the way a tutor would explain them at the board.

1Identify a translation
Getting started• translation
Question
Describe how $y = (x + 2)^2 - 5$ is obtained from $y = x^2$ .
Step-by-step solution
1. Step 1
  $(x + 2)$ inside the bracket: shift LEFT by 2.
2. Step 2
  $- 5$ outside: shift DOWN by 5.
3. Step 3
  Combined: shift left 2 and down 5. Vertex from $(0, 0)$ to $(-2, -5)$ .
Answer
Shift left 2 and down 5. New vertex $(-2, -5)$ .
2Identify a reflection
Building confidence• reflection
Question
Describe the transformation from $y = f(x)$ to $y = -f(x)$ , and apply to $y = x^3$ .
Step-by-step solution
1. Step 1
  Multiplying outside by $-1$ → reflection in the x-axis.
2. Step 2
  $y = x^3$ becomes $y = -x^3$ . The S-curve flips top-to-bottom.
Answer
Reflection in x-axis. $y = x^3$ becomes $y = -x^3$ .
3Combined transformations
Stretch• combined
Question
$y = 2(x - 3)^2 + 1$ — describe as a sequence of transformations of $y = x^2$ .
Step-by-step solution
1. Step 1
  $(x - 3)$ : shift RIGHT 3.
2. Step 2
  $\cdot 2$ outside: vertical stretch by 2 (parabola narrower).
3. Step 3
  $+ 1$ outside: shift UP 1.
4. Step 4
  Vertex at $(3, 1)$ , opens UP, narrower than standard.
Answer
Right 3, vertical stretch ×2, up 1. Vertex $(3, 1)$ .
4Equation from transformation
Stretch• equation
Question
A parabola has vertex $(-1, 4)$ and opens downward. Write the equation in vertex form.
Step-by-step solution
1. Step 1
  Vertex form: $y = a(x - h)^2 + k$ with $(h, k) = (-1, 4)$ .
2. Step 2
  Opens downward means $a < 0$ . Simplest choice: $a = -1$ .
3. Step 3
  Equation: $y = -(x + 1)^2 + 4$ .
Answer
$y = -(x + 1)^2 + 4$ (with any $a < 0$ ).

Key Definitions and Keywords — Transformation

Definitions to memorise and the exact keywords mark schemes credit for transformation answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Translation
Examiner keyword
A transformation that slides a graph without changing its shape.
Reflection
A transformation that flips a graph over a line.
Stretch (dilation)
A transformation that scales a graph by a factor in the x- or y-direction.
Vertex form
Examiner keyword
$y = a(x - h)^2 + k$ . Vertex is at $(h, k)$ .

Common Mistakes and Misconceptions — Transformation

The traps other students keep falling into on transformation questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Saying $f(x - 3)$ shifts the graph LEFT.
Why it happens
The minus 'feels' like negative direction.
How to avoid it
$f(x - 3)$ shifts the graph RIGHT by 3. Think of it as 'the input is $x - 3$ , which equals 0 when $x = 3$ ' — so the original $f(0)$ now appears at $x = 3$ .
✕Saying $f(2x)$ stretches horizontally by 2.
Why it happens
Multiplying suggests enlargement.
How to avoid it
$f(2x)$ COMPRESSES horizontally by factor 2 (i.e. half as wide). For horizontal stretch by 2, use $f(x/2)$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Transformation — frequently asked questions

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Transformation

Short Study Notes in the form of Slides

Short Study Notes — Transformation

Detailed Notes

What you’ll learn

How it’s examined

1Identify a translation

2Identify a reflection

3Combined transformations

4Equation from transformation

Translation

Reflection

Stretch (dilation)

Vertex form

✕Saying f(x−3)f(x - 3)f(x−3) shifts the graph LEFT.

✕Saying f(2x)f(2x)f(2x) stretches horizontally by 2.

Practice questions

Past paper style quiz

Assess your understanding

Transformation — frequently asked questions

✕Saying $f(x - 3)$ shifts the graph LEFT.

✕Saying $f(2x)$ stretches horizontally by 2.