Function Transformations — SL 2.6 Transformations of...

SECTION 1

Historical Context & Motivation

Long before graphing calculators existed, mathematicians needed efficient ways to understand how changing a formula would change its graph. Rather than re-plotting every single point from scratch, they discovered that certain algebraic changes — adding a constant, multiplying by a factor, or flipping a sign — always produce the same geometric effect on the curve. This insight, now called function transformation, lets you predict the shape and position of a graph without plotting dozens of points.

1637

Descartes Introduces Coordinate Geometry

René Descartes published La Géométrie, linking algebra to geometry by plotting equations on an x-y plane. This made it possible to visualize functions as curves.

1748

Euler Formalizes the Function Concept

Leonhard Euler defined a function as a rule mapping inputs to outputs. His notation f(x) is still the standard we use today, and it paved the way for analyzing how modifying the rule changes the output.

1800s

Systematic Study of Graph Behavior

Mathematicians like Fourier and Cauchy studied how adding constants, scaling, and reflecting functions affected their graphs, building the toolkit of transformations we learn in this topic.

1960s–Today

Graphing Technology & IB Curriculum

With graphing calculators and dynamic software like Desmos, students can now instantly see transformations in action. The IB curriculum (SL 2.6) formally teaches these ideas as essential tools for modeling real-world data.

The central question this topic addresses is simple but powerful: if you already know the graph of f(x), how can you quickly sketch the graph of a modified version like f(x) + 3, 2f(x), or f(−x) without recalculating every point? Understanding transformations means you only need to master a few parent functions and then adapt them to any situation.

SECTION 2

Core Principles & Definitions

Every function transformation falls into one of three families. A translation (or shift) slides the entire graph horizontally or vertically without changing its shape. A stretch (or compression) scales the graph by pulling it away from or squeezing it toward an axis. A reflection flips the graph across the x-axis or y-axis like a mirror. These three families, combined, account for all the transformations you need in SL 2.6.

Vertical Translation

Adding a constant d outside the function — f(x) + d — moves the graph up (d > 0) or down (d < 0) by |d| units.

Horizontal Translation

Replacing x with (x − c) inside the function — f(x − c) — moves the graph right (c > 0) or left (c < 0) by |c| units. Note the counter-intuitive sign!

Vertical Stretch / Compression

Multiplying outside: a × f(x). If |a| > 1 the graph stretches vertically; if 0 < |a| < 1 it compresses. The x-axis stays fixed.

Horizontal Stretch / Compression

Replacing x with bx inside the function — f(bx). If |b| > 1 the graph compresses horizontally; if 0 < |b| < 1 it stretches. This is the opposite of what many students expect.

Reflections

−f(x) reflects the graph in the x-axis (flip vertically). f(−x) reflects the graph in the y-axis (flip horizontally). Both simply negate coordinates.

✦ KEY TAKEAWAY

Think of a function's graph like a sticker on a transparent sheet. A translation slides the sheet without rotating or resizing the sticker. A stretch is like stretching or squishing the sheet (the sticker's shape changes). A reflection flips the sheet over. The key rule: changes outside the function affect the y-values (vertical), while changes inside the function affect the x-values (horizontal — and often do the opposite of what you'd guess).

SECTION 3

Visual Explanation — Translations

The diagram below shows the parent function f(x) = x² alongside three translated versions. Notice how each translation moves the entire parabola without altering its width or shape. The vertical shift moves every point the same number of units up or down, while the horizontal shift moves every point the same number of units left or right.

The dashed grey curve is the parent function f(x) = x². The cyan curve shows a vertical shift up by 1. The pink curve shows a horizontal shift left by 2, and the amber curve shows a horizontal shift right by 2.

A crucial pattern to notice: horizontal translations work in the opposite direction to the sign you see inside the brackets. Writing f(x − 2) moves the graph to the right, not the left. This is because when you replace x with (x − 2), every original output now requires an x-value that is 2 units larger to produce the same result. Vertical translations, on the other hand, are straightforward: f(x) + d shifts the graph in the direction of the sign of d.

SECTION 4

Mathematical Framework

The IB formula booklet presents transformations using a general template. Starting from a known function y = f(x), the transformed function can be written in a single expression that captures shifts, stretches, and reflections all at once.

GENERAL TRANSFORMATION

y = a × f(b(x − c)) + d

a = vertical stretch factor (if |a| > 1 stretch; if 0 < |a| < 1 compress; if a < 0 reflect in x-axis). b = horizontal stretch factor (graph compressed by factor 1/|b|; if b < 0 reflect in y-axis). c = horizontal translation (right if c > 0, left if c < 0). d = vertical translation (up if d > 0, down if d < 0).

VERTICAL TRANSLATION

y = f(x) + d

Every y-coordinate is increased by d. The graph moves up when d > 0 and down when d < 0.

HORIZONTAL TRANSLATION

y = f(x − c)

Every x-coordinate is increased by c. The graph moves right when c > 0 and left when c < 0. Remember the minus sign inside the bracket — it works opposite to what you might expect.

REFLECTIONS

y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis

For −f(x), every y-value is negated: points above the x-axis flip below, and vice versa. For f(−x), every x-value is negated: the graph is mirrored left ↔ right.

💡 IB Exam Tip

The IB often asks you to describe a transformation in words. Always state the type (translation, stretch, or reflection), the direction (horizontal or vertical), and the magnitude. For example: 'a horizontal translation of 3 units to the right' or 'a vertical stretch by a scale factor of 2'.

SECTION 5

Stretches, Compressions & Reflections in Detail

While translations simply slide a graph, stretches and reflections change its appearance more dramatically. A vertical stretch multiplies every y-coordinate by a scale factor, pulling the graph away from the x-axis. A horizontal compression squeezes the graph toward the y-axis. And reflections flip the graph across an axis like a mirror image. The diagram below uses f(x) = x² to illustrate these effects.

The amber curve (2f(x)) is twice as tall as the original — a vertical stretch by factor 2. The violet curve (f(2x)) is half as wide — a horizontal compression by factor ½. The red curve (−f(x)) is the parent flipped upside-down — a reflection in the x-axis.

Summary of stretches, compressions, and reflections

Transformation	Equation Form	Effect on Graph
Vertical stretch (factor a)	y = a × f(x), \|a\| > 1	Graph pulled away from x-axis; heights multiplied by a
Vertical compression (factor a)	y = a × f(x), 0 < \|a\| < 1	Graph pushed toward x-axis; heights multiplied by a
Horizontal compression (factor 1/b)	y = f(bx), \|b\| > 1	Graph squeezed toward y-axis; widths divided by \|b\|
Horizontal stretch (factor 1/b)	y = f(bx), 0 < \|b\| < 1	Graph stretched away from y-axis; widths multiplied by 1/\|b\|
Reflection in x-axis	y = −f(x)	All y-values negated; graph flips upside-down
Reflection in y-axis	y = f(−x)	All x-values negated; graph flips left-to-right

SECTION 6

Worked Example

Suppose you know the graph of f(x) = x². Describe the transformations needed to obtain the graph of g(x) = −2(x − 3)² + 5, and state the coordinates of the new vertex.

Transforming f(x) = x² into g(x) = −2(x − 3)² + 5

Step 1 — Match the General FormCompare g(x) = −2(x − 3)² + 5 with the template y = a × f(b(x − c)) + d. Here b = 1 (no horizontal stretch), so we identify a = −2, c = 3, and d = 5.

a = −2, c = 3, d = 5

Step 2 — Identify the Vertical Stretch and ReflectionSince a = −2, there are two effects combined. The factor |a| = 2 means a vertical stretch by a scale factor of 2 (every y-value is doubled). The negative sign means a reflection in the x-axis (the parabola opens downward instead of upward).

Vertical stretch × 2, then reflect in x-axis

Step 3 — Identify the Horizontal TranslationThe expression (x − 3) inside the function means translate 3 units to the right. Remember: (x − c) shifts right when c is positive.

Horizontal translation: 3 units right

Step 4 — Identify the Vertical TranslationThe +5 at the end means translate 5 units up.

Vertical translation: 5 units up

Step 5 — Find the New VertexThe original vertex of f(x) = x² is at (0, 0). After shifting right 3 and up 5, the new vertex is at (3, 5). The parabola opens downward because a is negative, and it is narrower than the standard parabola because |a| = 2 > 1.

Vertex = (3, 5); parabola opens downward

SECTION 7

Common Mistakes & Clarifications

Function transformations have a few well-known pitfalls that trip up even strong students. The table below highlights the most common mistakes alongside the correct interpretation. If you can avoid these errors, you'll handle most IB exam questions with confidence.

Top 5 transformation errors and how to fix them

Common Mistake	Why It's Wrong	Correct Approach
f(x + 3) shifts the graph right	The sign inside the bracket is opposite to the direction of movement	f(x + 3) shifts 3 units LEFT. To shift right, write f(x − 3).
f(2x) stretches the graph horizontally	Multiplying x by a number > 1 compresses, not stretches	f(2x) compresses horizontally by factor ½. To stretch, use f(½x).
Confusing −f(x) with f(−x)	One reflects vertically, the other horizontally — they are different transformations	−f(x) reflects in the x-axis. f(−x) reflects in the y-axis.
Applying transformations in the wrong order	Order matters when combining stretches, reflections, and translations	Apply stretches/reflections FIRST, then translations.
Forgetting to factor out b before identifying c	In f(2x − 6), the horizontal shift is NOT 6 — you must factor: f(2(x − 3))	Always write the inside as b(x − c) first. Here c = 3, so shift right 3.

✦ KEY TAKEAWAY

The 'inside vs. outside' rule is your best friend. Anything happening inside the function's brackets affects the x-direction and works in reverse — like reading a clock in a mirror, where 3 o'clock looks like it's on the left. Anything outside the function affects the y-direction and works exactly as you'd expect.

SECTION 8

Connection to Advanced Topics

Function transformations are not just an isolated topic — they form the foundation for many advanced ideas you'll encounter later in mathematics. The table below compares what you learn in SL 2.6 with how the same concepts appear at higher levels.

How SL 2.6 skills extend to advanced mathematics

SL 2.6 Concept	Advanced Extension	Where You'll See It
Vertical / horizontal translations	Vector translations in 2D and 3D geometry; phase shifts in trigonometric modeling	HL Geometry & Trigonometry, Physics (wave motion)
Vertical stretches	Amplitude changes in sinusoidal models; linear transformations of random variables in statistics	SL/HL Statistics, Physics (simple harmonic motion)
Horizontal stretches	Period changes in trigonometric functions; time-scaling in differential equations	HL Calculus, Signal Processing
Reflections	Symmetry analysis (even/odd functions); matrix transformations in linear algebra	HL Functions, University Linear Algebra
Combined transformations (general form)	Composition of functions; affine transformations in computer graphics	HL Functions, Computer Science

Mastering transformations now gives you a powerful mental toolkit. When you encounter sinusoidal models in Topic 3 or regression curves in Topic 4, you'll recognise the same patterns: a stretch here, a shift there, perhaps a reflection. The algebraic language of a, b, c, and d will reappear in every new function family you study.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain in your own words why f(x − 4) shifts the graph of f to the right rather than to the left, even though there is a minus sign.

PROBLEM 2 — BASIC CALCULATION

The point (2, 5) lies on the graph of y = f(x). Find the corresponding point on the graph of y = f(x) − 3.

PROBLEM 3 — INTERMEDIATE

The graph of y = f(x) passes through the point (4, −1). State the coordinates of the corresponding point on the graph of y = 3f(x − 2) + 1. Describe each transformation in order.

PROBLEM 4 — APPLIED

A company models its daily profit P (in thousands of dollars) with the function P(t), where t is hours after midnight. Due to a time zone change, the business day starts 2 hours earlier, and all profits are doubled. Write the transformed function in terms of P, and explain what each part of your expression means in context.

PROBLEM 5 — CRITICAL THINKING

Suppose g(x) = −f(2x + 6) − 4. (a) Rewrite the inside as 2(x − c) to correctly identify all transformations. (b) List every transformation in the order they should be applied. (c) If f has a maximum at (1, 8), find the coordinates of the corresponding point on g and state whether it is a maximum or minimum.

SUMMARY

Lesson Summary

Function transformations let you modify any parent function's graph using four parameters. A vertical translation (y = f(x) + d) slides the graph up or down. A horizontal translation (y = f(x − c)) slides it left or right — remembering that the direction is opposite to the sign. A vertical stretch (y = a × f(x)) scales y-values by factor |a|, while a horizontal stretch (y = f(bx)) compresses or stretches widths by factor 1/|b|. Reflections use negative signs: −f(x) flips across the x-axis, and f(−x) flips across the y-axis.

The general form y = a × f(b(x − c)) + d captures every SL 2.6 transformation. When applying multiple transformations, always perform stretches and reflections before translations. The inside-vs-outside rule is your compass: changes inside the brackets affect x (horizontally, in reverse), changes outside affect y (vertically, as expected). Master these patterns and you can transform any function's graph quickly and accurately on your IB exam.