IB Mathematics AA – Topic 2: Functions

Comprehensive Guide to Function Transformations

Introduction to Function Transformations

Function transformations are systematic changes applied to the graph of a function that alter its position, orientation, or shape without changing its fundamental character. Understanding transformations allows you to quickly sketch complex functions by starting with a basic "parent" function and applying a sequence of modifications.

Key concept: Given a parent function \(f(x)\), we can create new functions through transformations: translations (shifts), reflections (flips), stretches (elongations), and compressions (squeezes). Each transformation follows specific rules that determine how points on the original graph move to new positions.

Why transformations matter: Rather than plotting functions point-by-point, transformations provide a powerful shortcut. Recognizing that \(y = -(x-3)^2 + 5\) is a parabola shifted right 3, reflected over the x-axis, and shifted up 5 allows instant visualization and analysis.

In this guide: We'll master translations (horizontal and vertical shifts), reflections (flipping across axes), stretches and compressions (scaling), composite transformations (applying multiple changes), and develop the ability to predict and sketch transformed graphs confidently for IB exams.

1. Translations (Shifts)

What is a Translation?

A translation moves the entire graph of a function horizontally (left/right) or vertically (up/down) without changing its shape, size, or orientation. Every point on the graph shifts by the same amount in the same direction.

Translation Rules

Vertical Translation

\(y = f(x) + k\)

If \(k > 0\): shift up by \(k\) units
If \(k < 0\): shift down by \(|k|\) units
Point \((x, y)\) moves to \((x, y + k)\)

Example: \(y = x^2 + 3\) shifts the parabola up 3 units

Horizontal Translation

\(y = f(x - h)\)

If \(h > 0\): shift right by \(h\) units
If \(h < 0\): shift left by \(|h|\) units
Point \((x, y)\) moves to \((x + h, y)\)

Example: \(y = (x - 2)^2\) shifts the parabola right 2 units

⚠ Note: The sign is opposite to what you might expect!

⚠ Common Pitfalls:

Horizontal translation confusion: \(f(x - 3)\) shifts RIGHT 3, not left! The sign is opposite.
Order matters: \(f(x + 2)\) means replace \(x\) with \(x+2\), so left 2 units
Inside vs outside: Changes inside \(f(\cdots)\) affect x; outside affects y
Combined translations: Can do both: \(y = f(x - h) + k\)

Example 1: Translation of Functions

Problem: The function \(f(x) = x^2\) is transformed to \(g(x) = (x + 3)^2 - 4\)

(a) Describe the transformations

(b) State the new vertex

Solution:

(a) Transformations:

Compare \(g(x) = (x + 3)^2 - 4\) with \(f(x) = x^2\)

Horizontal: \((x + 3)\) means \(x - (-3)\), so shift left 3 units

Vertical: \(-4\) outside means shift down 4 units

Transformations: Left 3 units, Down 4 units

(b) New vertex:

Original vertex of \(f(x) = x^2\) is \((0, 0)\)

Apply transformations: left 3 → \(x = 0 - 3 = -3\)

Down 4 → \(y = 0 - 4 = -4\)

New vertex: \((-3, -4)\)

(c) Point transformation:

Original point: \((2, 4)\)

Left 3: \(x = 2 - 3 = -1\)

Down 4: \(y = 4 - 4 = 0\)

New point: \((-1, 0)\)

2. Reflections (Flips)

What is a Reflection?

A reflection flips the graph across a line (axis), creating a mirror image. The two most common reflections are across the x-axis and y-axis.

Reflection Rules:

Reflection in the x-axis

\(y = -f(x)\)

Negates all y-values
Point \((x, y)\) becomes \((x, -y)\)
Graph flips upside down

Example: \(y = -x^2\) is parabola opening downward (reflected over x-axis)

Reflection in the y-axis

\(y = f(-x)\)

Negates all x-values
Point \((x, y)\) becomes \((-x, y)\)
Graph flips horizontally (left-right)

Example: \(y = (-x)^2 = x^2\) (parabola unchanged—it's symmetric!)

Reflection in both axes

\(y = -f(-x)\)

Equivalent to 180° rotation about origin
Point \((x, y)\) becomes \((-x, -y)\)

💡 Reflection Tips:

Negative outside \(f\) → x-axis reflection (vertical flip)
Negative inside \(f\) → y-axis reflection (horizontal flip)
Check if function is even or odd—some look the same after reflection
Reflections change the function's increasing/decreasing behavior

3. Stretches and Compressions

Scaling Functions

Stretches and compressions change the shape of the graph by scaling it vertically (making it taller/shorter) or horizontally (making it wider/narrower).

Stretch/Compression Rules:

Vertical Stretch/Compression

\(y = a \cdot f(x)\) where \(a > 0\)

If \(a > 1\): vertical stretch by factor \(a\) (taller)
If \(0 < a < 1\): vertical compression by factor \(a\) (shorter)
Point \((x, y)\) becomes \((x, ay)\)
Multiplies all y-values by \(a\)

Example: \(y = 3x^2\) stretches parabola vertically by factor 3

Horizontal Stretch/Compression

\(y = f(bx)\) where \(b > 0\)

If \(b > 1\): horizontal compression by factor \(\frac{1}{b}\) (narrower)
If \(0 < b < 1\): horizontal stretch by factor \(\frac{1}{b}\) (wider)
Point \((x, y)\) becomes \(\left(\frac{x}{b}, y\right)\)
Effect is reciprocal of the coefficient!

Example: \(y = (2x)^2\) compresses parabola horizontally by factor \(\frac{1}{2}\)

⚠ The effect is the opposite of what the number suggests!

⚠ Critical Mistakes:

Horizontal stretch confusion: \(f(2x)\) compresses by \(\frac{1}{2}\), doesn't stretch by 2!
Sign handling: If coefficient is negative, include reflection as separate step
Order matters: \(2f(x)\) is different from \(f(2x)\)
Combined with translations: Usually do stretches/reflections before translations

Example 2: Stretches and Reflections

Problem: Given \(f(x) = \sqrt{x}\), describe the transformations and sketch:

(a) \(g(x) = -2f(x)\)

(b) \(h(x) = f(3x)\)

Solution:

(a) \(g(x) = -2f(x) = -2\sqrt{x}\)

Vertical stretch by 2: coefficient 2 outside \(f\)

Every y-value doubles: if point \((4, 2)\) on \(f\), becomes \((4, 4)\) after stretch

Reflection in x-axis: negative sign

After stretch, point \((4, 4)\) reflects to \((4, -4)\)

Combined: Vertical stretch by 2, then reflect in x-axis

Result: Function is "flipped" and "taller"

(b) \(h(x) = f(3x) = \sqrt{3x}\)

Horizontal compression by factor \(\frac{1}{3}\): coefficient 3 inside \(f\)

Effect is reciprocal: multiply x-values by \(\frac{1}{3}\)

If point \((9, 3)\) on \(f\) (since \(\sqrt{9} = 3\))

New x-coordinate: \(9 \times \frac{1}{3} = 3\)

Check: \(h(3) = \sqrt{3 \times 3} = \sqrt{9} = 3\) ✓

Transformation: Horizontal compression by factor \(\frac{1}{3}\)

Result: Graph is "narrower" (squeezed horizontally)

4. Composite Transformations

Combining Multiple Transformations

Real-world problems often require applying multiple transformations to a single function. The order in which you apply them matters!

Standard Order of Transformations:

For \(y = a \cdot f(b(x - h)) + k\):

Horizontal stretch/compression: factor \(\frac{1}{b}\) (inside, with x)
Horizontal translation: shift right \(h\) (inside, with x)
Vertical stretch/compression: factor \(a\) (outside)
Reflection: if \(a < 0\) or \(b < 0\)
Vertical translation: shift up \(k\) (outside, last)

Memory tip: Inside transformations (horizontal) before outside transformations (vertical)

Reading Transformations from Equations

Systematic Approach:

Identify the parent function
Look inside the function for horizontal changes
Look outside the function for vertical changes
Check for negative signs (reflections)
Identify coefficients (stretches/compressions)
Apply transformations in correct order

Example 3: Complete Transformation Analysis (IB-Style)

Problem: The parent function is \(f(x) = |x|\). The transformed function is:

\(g(x) = -2f\left(\frac{1}{3}(x + 6)\right) - 5\)

(a) List all transformations in order

(b) If point \((3, 3)\) is on \(f(x)\), where is it on \(g(x)\)?

Solution:

(a) Identifying transformations:

Rewrite: \(g(x) = -2f\left(\frac{1}{3}x + 2\right) - 5\)

Step 1: Horizontal stretch

Inside: \(\frac{1}{3}x\) means coefficient \(b = \frac{1}{3}\)

Horizontal stretch by factor \(\frac{1}{b} = 3\) (wider)

Step 2: Horizontal translation

Inside: \(\frac{1}{3}(x + 6) = \frac{1}{3}x + 2\)

This is \(f\left(\frac{1}{3}(x - (-6))\right)\), so left 6 units

Step 3: Vertical stretch

Outside: coefficient 2

Vertical stretch by factor 2 (taller)

Step 4: Reflection in x-axis

Outside: negative sign

Reflect over x-axis (flip upside down)

Step 5: Vertical translation

Outside: \(-5\)

Shift down 5 units

Complete list:
1. Horizontal stretch by 3
2. Horizontal translation left 6
3. Vertical stretch by 2
4. Reflection in x-axis
5. Vertical translation down 5

(b) Point transformation:

Start: \((3, 3)\)

Horizontal stretch ×3: \((9, 3)\)

Left 6: \((9 - 6, 3) = (3, 3)\)

Vertical stretch ×2: \((3, 6)\)

Reflect in x-axis: \((3, -6)\)

Down 5: \((3, -11)\)

New point: \((3, -11)\)

(c) New vertex:

Original vertex of \(f(x) = |x|\) is \((0, 0)\)

Horizontal stretch ×3: \((0, 0)\) (no change—on axis)

Left 6: \((-6, 0)\)

Vertical stretch ×2: \((-6, 0)\) (no change—y already 0)

Reflect in x-axis: \((-6, 0)\) (no change—y already 0)

Down 5: \((-6, -5)\)

New vertex: \((-6, -5)\)

📋 Transformation Quick Reference

Transformation	Notation	Effect	Point Change
Vertical Translation	\(f(x) + k\)	Up \(k\) (if \(k > 0\))	\((x, y+k)\)
Horizontal Translation	\(f(x - h)\)	Right \(h\) (if \(h > 0\))	\((x+h, y)\)
Reflection (x-axis)	\(-f(x)\)	Flip vertically	\((x, -y)\)
Reflection (y-axis)	\(f(-x)\)	Flip horizontally	\((-x, y)\)
Vertical Stretch	\(a \cdot f(x)\), \(a>1\)	Taller by \(a\)	\((x, ay)\)
Horizontal Compression	\(f(bx)\), \(b>1\)	Narrower by \(\frac{1}{b}\)	\((\frac{x}{b}, y)\)

🎯 IB Exam Strategy

Common Question Types:

"Describe the transformations": Work systematically inside to outside
"Sketch the transformed graph": Transform key points then connect
"Find the equation": Build up from parent function applying each transformation
"Where does point (a,b) move?": Apply each transformation to coordinates

Key Reminders:

Horizontal changes have opposite effect: \(f(x-3)\) goes right, not left
Horizontal stretch by 2 means \(f(\frac{x}{2})\), not \(f(2x)\)
Order matters: do inside transformations before outside
Always identify parent function first
Use GDC to verify your sketch

🎉 Master Function Transformations!

Understanding transformations gives you the power to visualize and sketch complex functions instantly. Rather than plotting points, you can transform familiar parent functions through systematic steps—a crucial skill for IB exams and higher mathematics!

Key Success Factors:

✓ Inside function = horizontal (opposite effect!)
✓ Outside function = vertical (direct effect)
✓ Order: horizontal first, then vertical
✓ \(f(x-h)+k\): right \(h\), up \(k\)
✓ Negative outside = x-axis reflection
✓ Practice transforming key points systematically

Think Systematically • Apply in Order • Transform Points

Master transformations and visualize any function! 🚀