Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)
Published: October 2025
Category: Algebra | Graphs & Functions

🔹 What Are Function Transformations?

A transformation of a function changes its shape, position, or orientation on a graph — without altering its fundamental nature.

Every transformation can be understood as a modification of the base function f(x)f(x)f(x).

For example:

• y=f(x)+ky = f(x) + ky=f(x)+k → vertical shift
• y=f(x−h)y = f(x – h)y=f(x−h) → horizontal shift
• y=−f(x)y = -f(x)y=−f(x) → reflection about x-axis
• y=af(x)y = a f(x)y=af(x) → vertical stretch or compression

Think of transformations as “how functions move and morph on the graph paper.”

🔹 The Four Core Transformations

Let’s break them down with meaning and examples.

1️⃣ Vertical Shifts (Up/Down)

Adding or subtracting a constant outside the function moves the graph up or down. y=f(x)+ky = f(x) + ky=f(x)+k

• If k>0k > 0k>0: shift up by kkk
• If k<0k < 0k<0: shift down by ∣k∣|k|∣k∣

🧩 Example: y=x2+3y = x^2 + 3y=x2+3

is the graph of y=x2y = x^2y=x2 shifted up 3 units.

2️⃣ Horizontal Shifts (Left/Right)

Adding or subtracting inside the function argument moves it left or right. y=f(x−h)y = f(x – h)y=f(x−h)

• If h>0h > 0h>0: shift right by hhh
• If h<0h < 0h<0: shift left by ∣h∣|h|∣h∣

🧩 Example: y=(x−2)2y = (x – 2)^2y=(x−2)2

is the graph of y=x2y = x^2y=x2 shifted 2 units right.

3️⃣ Reflections (Flipping the Graph)

Flips the graph across an axis.

• y=−f(x)y = -f(x)y=−f(x): reflect about the x-axis
• y=f(−x)y = f(-x)y=f(−x): reflect about the y-axis

🧩 Example: y=−∣x∣y = -|x|y=−∣x∣

is the reflection of y=∣x∣y = |x|y=∣x∣ downward across the x-axis.

4️⃣ Stretches and Compressions

Multiplying either the output or input by a constant changes the steepness or width of the graph.

(a) Vertical Stretch/Compression

y=af(x)y = a f(x)y=af(x)

• ∣a∣>1|a| > 1∣a∣>1: vertical stretch
• 0<∣a∣<10 < |a| < 10<∣a∣<1: vertical compression

🧩 Example: y=2x2y = 2x^2y=2×2

is narrower (stretched vertically) than y=x2y = x^2y=x2.

(b) Horizontal Stretch/Compression

y=f(bx)y = f(bx)y=f(bx)

• ∣b∣>1|b| > 1∣b∣>1: compressed horizontally (narrower)
• 0<∣b∣<10 < |b| < 10<∣b∣<1: stretched horizontally (wider)

🧩 Example: y=(2x)2=4x2y = (2x)^2 = 4x^2y=(2x)2=4×2

is narrower than y=x2y = x^2y=x2.

🔹 Combined Transformations

Most exam questions (IB, A Level, AP) mix multiple transformations.
The key is to apply transformations in the correct order — usually inside → outside (horizontal → vertical).

🧩 Example: y=−2(x−1)2+3y = -2(x – 1)^2 + 3y=−2(x−1)2+3

Step-by-step:
1️⃣ Shift right 1 unit → (x−1)(x – 1)(x−1)
2️⃣ Stretch vertically by factor 2 → 2(x−1)22(x – 1)^22(x−1)2
3️⃣ Reflect across x-axis → −2(x−1)2-2(x – 1)^2−2(x−1)2
4️⃣ Shift up 3 units → +3+3+3

✅ Result: a parabola opening downward, vertex at (1, 3), narrower than y=x2y = x^2y=x2.

🔹 Symmetry & Invariance

Transformations preserve symmetry if applied carefully:

• Reflection about y-axis preserves even functions (f(x)=f(−x)f(x) = f(-x)f(x)=f(−x))
• Reflection about x-axis flips the sign of all outputs
• Translations preserve overall shape but not symmetry center

🔹 Common Mistakes

1. ❌ Mixing up inside vs outside shifts (horizontal vs vertical).
2. ❌ Forgetting sign reversal — f(x−h)f(x – h)f(x−h) moves right, not left.
3. ❌ Applying transformations in the wrong order.
4. ❌ Ignoring negative scaling factors (they cause reflections).

🔹 Advanced Insight — Function Composition

All transformations can be written as compositions: y=af(b(x−h))+ky = a f(b(x – h)) + ky=af(b(x−h))+k

Where:

• hhh: horizontal shift
• bbb: horizontal stretch/compression
• aaa: vertical stretch/compression
• kkk: vertical shift

This unified form helps in graph sketching, transformations, and inverse function questions.

🔹 Real-World Applications

• Physics: wave transformations (phase shift, amplitude, frequency)
• Economics: demand/supply function scaling
• Computer graphics: geometric transformations
• Math exams: core for IB, IGCSE, A Level, and AP

🌟 Why This Topic Matters

Understanding transformations builds graphical intuition — the ability to visualize equations before plotting them.
It turns algebra into geometry — a crucial skill for competitive exams like STEP and MAT, where interpretation matters more than formula.

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✅ Master transformations through geometry & animation,
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