IB Mathematics AI AHL Transformations of graphs MAI Study Notes - New Syllabus

IB Mathematics AI AHL Transformations of graphs MAI Study Notes

LEARNING OBJECTIVE

Transformations of graphs.

Key Concepts:

Translations of Graphs
Reflections of Graphs
Stretches of Graphs
Composite Transformations of Graphs

Transformations describe how a function’s graph changes in position or shape. These include translations (shifts), reflections (flips), and stretches/compressions (scaling). The core function (often called the parent function) stays the same in type but is adjusted visually.

Vertical Translations

These move the graph up or down without altering its shape.

General Form:

$
y = f(x) + a
$

If $a > 0$: moves up by $a$ units.
If $a < 0$: moves down by $|a|$ units.

Question

Consider the function $ f(x) = x^2 $)

Describe the transformation: $ f(x) + a $
Describe the transformation: $ f(x) – a $

▶️Answer/Explanation

Solution:

Transformation: Vertical translation a units up
( g(x) = x^2 + 2 \) → The graph moves up 2 units
Transformation: Vertical translation a units down
$ g(x) = x^2 – 2 $ → The graph moves down 2 units

Example:

$
f(x) = x^2 \quad \Rightarrow \quad g(x) = x^2 + 3
$

Graph of $f(x)$ shifts up 3 units.

Horizontal Translations

These move the graph left or right.

General Form:

$
y = f(x – a)
$

If $a > 0$: moves right by $a$ units.
If $a < 0$: moves left by $|a|$ units.

Example:

$
f(x) = x^2 \quad \Rightarrow \quad g(x) = (x – 2)^2
$

Graph moves right 2 units.

Note: The sign inside the brackets is opposite the direction.

Question

Consider the function $ f(x) = x^2 $)

Describe the transformation: $ f(x + a) $
Describe the transformation: $ f(x – a) $

▶️Answer/Explanation

Solutions:

Transformation: Horizontal translation a units to the left
$ g(x) = (x + 2)^2 $ → The graph moves 2 units left
Transformation: Horizontal translation a units to the right
$ g(x) = (x – 2)^2 $ → The graph moves 2 units right

Reflections flip the graph over an axis.

Reflection in the x-axis:

$
y = -f(x)
$

Flips vertically—every y-value becomes its opposite.

Reflection in the y-axis:

$
y = f(-x)
$

Flips horizontally—every x-value is mirrored.

Question

Consider the function $ f(x) = x^2 + 9 $):

Describe the transformation: $ -f(x) $
Describe the transformation: $ f(-x) $

▶️Answer/Explanation

Solution:

Transformation: Reflection in the x-axis
$ g(x) = -(x^2 + 9) $ → The graph is flipped upside down
Transformation: Reflection in the y-axis
$ g(x) = (-x)^2 + 9 $ → The graph is reflected over the y-axis (left ↔ right)

Vertical Stretch and Compression

It affects the height of the graph.

General Form:

$
y = a \cdot f(x)
$

$|a| > 1$: stretched vertically.
$0 < |a| < 1$: compressed vertically.
$a < 0$: reflected in the x-axis.

Example:

$
f(x) = x^2 \Rightarrow g(x) = 3x^2
$

The graph is stretched by a factor of 3.

Horizontal Stretch and Compression

It affects how “wide” the graph appears.

General Form:

$
y = f(bx)
$

$|b| > 1$: compressed horizontally.
$0 < |b| < 1$: stretched horizontally.
$b < 0$: reflected in the y-axis.

Example:

$
f(x) = x^2 \Rightarrow g(x) = f(2x) = (2x)^2 = 4x^2
$

The graph is compressed horizontally by factor $\frac{1}{2}$.

Common Confusion: Inside the function → horizontal effect. Outside the function → vertical effect.

Question:

Given the function $ f(x) = x^2 $, consider the transformation

$ g(x) = -2 \cdot f(0.5x) $

What are the effects of this transformation on the graph of $ f(x) $? Describe each transformation step (vertical/horizontal stretch/compression and reflection), and write the simplified form of $ g(x) $.

▶️Answer/Explanation

Solution:

$ g(x) = -2 \cdot f(0.5x) = -2 \cdot (0.5x)^2 $

$ g(x) = -2 \cdot 0.25x^2 = -0.5x^2 $

Transformations:

Horizontal Stretch:
The $ 0.5x $ is inside the function.
Since $ |0.5| < 1 $, the graph is stretched horizontally by a factor of $$ \frac{1}{0.5} = 2. $$
Vertical Stretch:
The coefficient 2 is outside the function.
Since $ |2| > 1 $, the graph is stretched vertically by a factor of 2.
Reflection in x-axis:
The negative sign in front of the function means the graph is reflected over the x-axis.

The graph of $ f(x) = x^2 $ becomes wider (horizontal stretch by 2), steeper (vertical stretch by 2), and opens downward (reflected in the x-axis) to form the new function:

$ g(x) = -0.5x^2 $

Transformations often relate to composite functions . You can view transformations as function compositions.

Example:

$
g(x) = 2(x – 3)^2 + 1 = h(f(x)), \text{ where:}
$

$f(x) = x – 3$
$h(x) = 2x^2 + 1$

This breaks the transformation into smaller steps using function composition.

Using Technology

Use graphing tools like Desmos or GeoGebra to experiment with transformations:

Add sliders for each parameter (e.g., shift, scale).
Observe real-time changes to the graph.
Compare multiple versions simultaneously.

Sketching Strategy

When sketching transformed graphs:

1. Start with the basic shape of the parent function.
2. Apply reflections first.
3. Follow with stretches/compressions.
4. Then apply translations (shifts).
5. Plot key points to guide the shape.
6. Don’t stress over exact scale—focus on structure and movement.

Question:

The parent function $ f(x) = \sqrt{x} $ undergoes the following transformations to become the function $ g(x) = -2\sqrt{x – 3} + 1 $.

Using the sketching strategy provided:

Describe the sequence of transformations applied to $ f(x) $.
Sketch the graph of $ g(x) $, showing at least three key points and identifying any changes in domain and range.

▶️Answer/Explanation

Solution:

$ f(x) = \sqrt{x} $

Inside the function: $ \sqrt{x – 3} $
Translation to the right by 3 units → Horizontal shift

Coefficient outside: $ -2\sqrt{x – 3} $
Vertical stretch by a factor of 2
Reflection over the x-axis (because of the negative sign)

Outside +1: $ -2\sqrt{x – 3} + 1 $
Translation upward by 1 unit

Transformations: