📐 Concept diagram

03-02 - Transformations of Functions

Phase: 3 | Subject: 03-02 Prerequisites: 03-01-functions.md (function notation, graphs) Next subject: 03-03-polynomial-functions.md

Learning Objectives

By the end of this subject, you will be able to:

1. Apply vertical and horizontal shifts to functions
2. Apply vertical and horizontal stretches/compressions
3. Apply reflections across the axes
4. Combine multiple transformations in the correct order
5. Write the equation of a transformed function from a graph

Core Content

The General Transformation Formula

For a base function f(x), the transformed function g(x) is:

$g(x) = a · f(b(x - c)) + d
$

Common Pitfall: Inside vs Outside

Inside the function (with x): horizontal transformation - f(x + 3) shifts LEFT 3 (opposite of what you might expect!) - f(2x) compresses HORIZONTALLY by factor 2

Outside the function (with the output): vertical transformation - f(x) + 3 shifts UP 3 - 2f(x) stretches VERTICALLY by factor 2

Memory rule: "Inside is opposite, outside is same."

Order of Transformations

When multiple transformations are applied, the order matters:

1. Horizontal shift (inside)
2. Horizontal stretch/compression (inside)
3. Reflection across y-axis (inside, if b < 0)
4. Reflection across x-axis (outside, if a < 0)
5. Vertical stretch/compression (outside)
6. Vertical shift (outside)

Example: y = -2f(3(x - 1)) + 4 applied to f(x) = x²:

1. Start: y = x²
2. Horizontal shift right 1: y = (x - 1)²
3. Horizontal compress by 3: y = (3x - 3)² or y = (3(x-1))²
4. Reflect across x-axis: y = -(3x - 3)²
5. Vertical stretch by 2: y = -2(3x - 3)²
6. Vertical shift up 4: y = -2(3x - 3)² + 4

Specific Transformations

Vertical Shift: f(x) + k

• k > 0: shift UP k units
• k < 0: shift DOWN |k| units

Horizontal Shift: f(x - h)

• h > 0: shift RIGHT h units
• h < 0: shift LEFT |h| units

Vertical Stretch: a·f(x)

• |a| > 1: stretch vertically (taller)
• 0 < |a| < 1: compress vertically (shorter)
• a < 0: also reflect across x-axis

Horizontal Stretch: f(bx)

• |b| > 1: compress horizontally (narrower)
• 0 < |b| < 1: stretch horizontally (wider)
• b < 0: also reflect across y-axis

Key Terms

• Inside the function
• Outside the function

Worked Examples

Example 1: Describe transformations

g(x) = 3f(x + 2) - 1

1. Horizontal shift LEFT 2 (inside: x + 2)
2. Vertical stretch by 3
3. Vertical shift DOWN 1

Example 2: Find equation from graph

Graph of y = √x transformed: - Shifted right 3 - Stretched vertically by 2 - Shifted down 1

g(x) = 2√(x - 3) - 1

Example 3: Combined transformations

y = -½f(2x + 4) + 3 applied to f(x) = x²

1. Rewrite: y = -½f(2(x + 2)) + 3
2. Horizontal shift LEFT 2
3. Horizontal compress by 2
4. Vertical compress by ½
5. Reflect across x-axis
6. Vertical shift UP 3

Quiz

Q1: What does the concept of Inside the function primarily refer to in this subject?

A) A visual representation of Inside the function B) A historical anecdote about Inside the function C) The definition and application of Inside the function D) A computational error related to Inside the function

Correct: C)

• If you chose A: This is incorrect. Inside the function is defined as: the definition and application of inside the function. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Inside the function is defined as: the definition and application of inside the function. The other options describe different aspects that are not the primary focus.
• If you chose C: Inside the function is defined as: the definition and application of inside the function. The other options describe different aspects that are not the primary focus. Correct!
• If you chose D: This is incorrect. Inside the function is defined as: the definition and application of inside the function. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Outside the function?

A) It is primarily a historical notation system B) It is used only in advanced research contexts C) It is used to outside the function in mathematical analysis D) It replaces all other methods in this domain

Correct: C)

• If you chose A: This is incorrect. Outside the function serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Outside the function serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Outside the function serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Outside the function serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about The General Transformation Formula is TRUE?

A) The General Transformation Formula is mentioned only as a historical footnote B) The General Transformation Formula is an advanced topic beyond this subject's scope C) The General Transformation Formula is not related to this subject D) The General Transformation Formula is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. The General Transformation Formula is a fundamental concept covered in this subject. This subject covers The General Transformation Formula as part of its core content.
• If you chose B: This is incorrect. The General Transformation Formula is a fundamental concept covered in this subject. This subject covers The General Transformation Formula as part of its core content.
• If you chose C: This is incorrect. The General Transformation Formula is a fundamental concept covered in this subject. This subject covers The General Transformation Formula as part of its core content.
• If you chose D: The General Transformation Formula is a fundamental concept covered in this subject. This subject covers The General Transformation Formula as part of its core content. Correct!

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) A different result from a common mistake C) The inverse of the correct answer D) ** Shift right 4, then shift up 2.

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is ** Shift right 4, then shift up 2.. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is ** Shift right 4, then shift up 2.. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is ** Shift right 4, then shift up 2.. The other options represent common errors.
• If you chose D: The worked examples show that the result is ** Shift right 4, then shift up 2.. The other options represent common errors. Correct!

Q5: How are The General Transformation Formula and Common Pitfall: Inside Vs Outside related?

A) The General Transformation Formula is a special case of Common Pitfall: Inside Vs Outside B) The General Transformation Formula is the inverse of Common Pitfall: Inside Vs Outside C) The General Transformation Formula and Common Pitfall: Inside Vs Outside are completely unrelated topics D) The General Transformation Formula and Common Pitfall: Inside Vs Outside are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both The General Transformation Formula and Common Pitfall: Inside Vs Outside are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both The General Transformation Formula and Common Pitfall: Inside Vs Outside are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both The General Transformation Formula and Common Pitfall: Inside Vs Outside are covered in this subject as interconnected topics.
• If you chose D: Both The General Transformation Formula and Common Pitfall: Inside Vs Outside are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Order Of Transformations?

A) The main error with Order Of Transformations is using it when it is not needed B) Order Of Transformations has no common misconceptions C) Order Of Transformations is always computed the same way in all contexts D) A common mistake is confusing Order Of Transformations with a similar concept

Correct: D)

• If you chose A: This is incorrect. Students often confuse Order Of Transformations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Order Of Transformations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Order Of Transformations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: Students often confuse Order Of Transformations with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!

Q7: When should you apply Specific Transformations?

A) Use Specific Transformations only in pure mathematics contexts B) Specific Transformations is not practically useful C) Avoid Specific Transformations unless explicitly instructed D) Apply Specific Transformations to solve problems in this subject's domain

Correct: D)

• If you chose A: This is incorrect. Specific Transformations is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Specific Transformations is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Specific Transformations is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: Specific Transformations is a practical tool used throughout this subject to solve relevant problems. Correct!

Practice Problems

1. Describe g(x) = f(x - 4) + 2 in words. Answer: Shift right 4, then shift up 2.

2. g(x) = -3f(2x) is applied to f(x) = x². Result? Answer: Reflect across x-axis, vertical stretch by 3, horizontal compress by 2. y = -3(2x)² = -12x².

3. If f(x) = √x, what is g(x) = f(x + 3) - 2? Answer: Shift left 3, down 2. g(x) = √(x + 3) - 2.

4. g(x) = 2f(3x) applied to f(x) = sin(x). Result? Answer: Horizontal compress by 3, vertical stretch by 2. g(x) = 2sin(3x).

5. Describe: y = f(-x) Answer: Reflect across y-axis (horizontal reflection).

Summary

Key takeaways:

• g(x) = a·f(b(x - c)) + d
• Inside (x - c): horizontal shift — OPPOSITE direction
• Outside (+ d): vertical shift — SAME direction
• |a| > 1: vertical stretch; 0 < |a| < 1: vertical compression
• |b| > 1: horizontal compression; 0 < |b| < 1: horizontal stretch
• a < 0 or b < 0: reflection
• Order: horizontal first, then vertical

Pitfalls

• Getting horizontal shifts backwards: f(x + 3) shifts LEFT, not right. The sign inside the function is opposite to the direction of the shift. This is the single most common transformation error — the "inside is opposite" rule takes practice to internalise.
• Applying transformations in the wrong order: Transformations are not commutative. Horizontal transformations should be applied first (shift, then stretch/compress), followed by vertical ones (reflect, stretch/compress, then shift). A different order produces a different graph.
• Forgetting to factor before identifying transformations: f(2x + 6) must be rewritten as f(2(x + 3)). Without factoring, you might think the horizontal shift is 6 when it's actually 3, or the compress factor is unclear.
• Misreading stretch vs. compression: For vertical transformations (outside): |a| > 1 stretches, 0 < |a| < 1 compresses. For horizontal transformations (inside): |b| > 1 compresses, 0 < |b| < 1 stretches. Inside is opposite — larger number means narrower graph.
• Confusing horizontal and vertical reflections: f(-x) reflects across the y-axis (horizontal). -f(x) reflects across the x-axis (vertical). A negative OUTSIDE flips the y-values; a negative INSIDE flips the x-values.

Next Steps

Next up: 03-03-polynomial-functions.md