🎯⭐ INTERACTIVE LESSON

Transformations

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Transformations - Complete Interactive Lesson

Part 1: Translations

📐 Translations (Shifts)

Part 1 of 7

Vertical Shifts

$y = f(x) + k$

$k > 0$	Shift up $k$ units
$k < 0$	Shift down $

Horizontal Shifts

$y = f(x - h)$

$h > 0$	Shift right $h$ units
$h < 0$	Shift left $

⚠️ Horizontal shifts work opposite to what you might expect! $f(x-3)$ shifts right 3, not left.

Combined Example

$y = (x-2)^2 + 5$ : Take $y=x^2$ , shift right , up . Vertex: .

📝 Worked Examples

Example 1: $y = \sqrt{x+4} - 3$

Start with $y = \sqrt{}$ .

🧠 The General Translation

$y - k = f(x - h) \quad \Leftrightarrow \quad y = f(x-h)+k$

Translations Quiz 🎯

Translation Practice 🧮

1) The graph of $y=x^2$ is shifted left 3 and up 2. New equation vertex $h$ = ?

2) Same: vertex $k$ = ?

3) $y = (x + 1)^{3} - 4$ has inflection point at = ?

Translation Concepts 🔽

Exit Quiz ✅

Part 2: Reflections

🔄 Reflections

Part 2 of 7

Reflection Over the $x$ -axis

$y = -f(x)$

Negate the output: flip the graph upside down.

Reflection Over the $y$ -axis

$y = f(-x)$

Part 3: Stretches & Compressions

📏 Stretches & Compressions

Part 3 of 7

Vertical Stretch/Compression

$y = a \cdot f(x)$

| $|a| > 1$ | Vertical stretch by factor $∣ a ∣$ | |:----------|:------------------------------------| | | Vertical by factor | | | Also reflects over -axis |

Part 4: Combined Transformations

🔗 Combining Multiple Transformations

Part 4 of 7

The General Form

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

Order of Operations

Apply in this order:

Horizontal shift (inside):

Part 5: Piecewise Functions

📚 Parent Functions Gallery

Part 5 of 7

The Essential Toolkit

Every function you transform begins as one of these parent functions.

Function	Equation	Key Features
Linear	$y = x$	Slope 1, through origin
Quadratic	$y = x^2$

Part 6: Problem-Solving Workshop

📐 Writing Equations from Graphs

Part 6 of 7

The Reverse Problem

Instead of transforming a parent → graph, we now go from graph → equation.

Step-by-Step Strategy

Identify the parent — What shape is it? (parabola, V, S-curve, etc.)
Locate the key point — Vertex, center, inflection point → gives $(h, k)$
Check orientation — Is it flipped? → sign of $a$
Find the scale — Plug in a visible point to solve for $|a|$
Verify — Test another point if possible

Part 7: Review & Applications

🏆 Transformations — Full Synthesis

Part 7 of 7

Master Checklist

Transformation	Formula	Effect
Vertical shift	$f(x)+k$	Up ( $k>0$ ) or down ( $k <$ )

Replace $x$ with $x+4$ : shift left $4$
Subtract $3$ : shift down $3$

Starting point moves from $(0,0)$ to $(-4, -3)$ .

Example 2: $y = |x-1| + 2$

Start with $y = |x|$ .

Replace $x$ with $x-1$ : shift right $1$
Add $2$ : shift up $2$

Vertex moves from $(0,0)$ to $(1, 2)$ .

Why Horizontal Shifts Are "Backwards"

$f(x-3) = 0$ when $x-3 = 0$ , i.e., $x = 3$ . The zero moved right by $3$ .

The $x$ -value must be $3$ more to produce the same result → the graph shifts right.

Every point $(a, b)$ on $y=f(x)$ moves to $(a+h, b+k)$ .

Translating Key Points

For $y = (x-3)^3 + 1$ (parent: $y = x^3$ ):

Parent point	Translated point
$(-1, -1)$	$(2, 0)$
$(0, 0)$	$(3, 1)$
$(1, 1)$	$(4, 2)$

Effect on Domain and Range

If $f$ has domain $[a, b]$ and range $[c, d]$ :

$y = f(x-h)+k$ has domain $[a+h, b+h]$ and range $[c+k, d+k]$ .

y = (x + 1)^{3} - 4

Negate the input: flip the graph left-right.

Transformation	Effect	Example
$-f(x)$	Reflect over $x$ -axis	$-x^2$ : opens down
$f(-x)$	Reflect over $y$ -axis	$(-x)^3 = -x^3$
$-f(-x)$	Reflect over both (= rotate $180°$ )	Origin symmetry

📝 Examples

Example 1: $y = -\sqrt{x}$

Start with $y = \sqrt{x}$ (half-parabola in Q1).

$-f(x)$ : reflect over $x$ -axis → now in Q4.

Points: $(0,0) \to (0,0)$ , $(4,2) \to (4,-2)$ , .

Example 2: $y = \sqrt{-x}$

$f(-x)$ : reflect over $y$ -axis → now in Q2.

Points: $(0,0)\to(0,0)$ , $(4,2)\to(-4,2)$ .

Even and Odd Functions

Even: $f(-x) = f(x)$ → symmetric about $y$ -axis (e.g., $x^2, \cos x$ )

💡 Reflecting an even function over the $y$ -axis gives the same graph!

🔀 Combining Reflections with Shifts

Order matters! Apply transformations in the correct sequence.

Example: $y = -|x-2|+3$

Start with $y = |x|$
Shift right 2: $y = |x-2|$
Reflect over $x$ -axis: $y = -|x-2|$
Shift up 3: $y = -|x-2|+3$

Vertex: $(2, 3)$ , opening downward.

Example: $y = (-x)^3 + 1 = -x^3+1$

Start with $y = x^3$
Reflect over $y$ -axis: $y = (-x)^3 = -x^3$

💡 For odd functions, reflecting over the $y$ -axis is the same as reflecting over the $x$ -axis!

Reflections Quiz 🎯

Reflections Practice 🧮

The point $(4, 7)$ is on $y = f(x)$ .

1) On $y = -f(x)$ , this becomes $(4,$ ? $)$

2) On $y = f(-x)$ , this becomes $($ ? $, 7)$

3) Is $f(x) = x^4 + x^2$ even, odd, or neither? (Type "even", "odd", or "neither")

Reflection Concepts 🔽

Horizontal Stretch/Compression

| $|b| > 1$ | Horizontal compression by factor $\frac{1}{|b|}$ | |:----------|:-----------------------------------------------------| | $0 < |b| < 1$ | Horizontal stretch by factor $\frac{1}{|b|}$ |

⚠️ Horizontal scaling is reciprocal: $f(2x)$ compresses by half, not stretches by 2!

📝 Examples

Example 1: $y = 3\sin x$

Vertical stretch by 3. Amplitude changes from 1 to 3.

Points: $(\pi/2, 1) \to (\pi/2, 3)$ .

Example 2: $y = \sin(2x)$

Horizontal compression by $1/2$ . Period changes from $2\pi$ to $\pi$ .

Points: $(\pi/2, 1) \to (\pi/4, 1)$ .

Example 3: $y = \frac{1}{2}x^2$

Vertical compression by $1/2$ . The parabola is "wider."

Points: $(2, 4) \to (2, 2)$ , $(4, 16) \to (4, 8)$ .

Key Insight

Vertical changes multiply $y$ -values.

Horizontal changes divide $x$ -values by $b$ (or multiply by $1/b$ ).

🔄 Effect on Period & Amplitude

For trig functions $y = A\sin(Bx)$ :

Amplitude $= |A|$ (vertical stretch)
Period $= \frac{2\pi}{|B|}$ (horizontal compression)

Example: $y = 4\cos(3x)$

Amplitude: $4$ , Period: $\frac{2\pi}{3}$

For General Functions

Original Feature	After $y = af(bx)$
Point $(x, y)$	$(x/b, ay)$

Stretches Quiz 🎯

Stretch Calculations 🧮

1) $y = 2f(x)$ : the point $(3, 5)$ becomes $(3,$ ? $)$

2) $y = f(4x)$ : the point $(8, 5)$ becomes $($ ? $, 5)$

3) The period of $y = \sin(4x)$ : $\frac{2\pi}{?}$ . Enter the number.

Stretch Concepts 🔽

Horizontal stretch/reflect (inside): multiply by

b

Vertical stretch/reflect (outside): multiply by

a

Vertical shift (outside): add

k

💡 Inside transformations affect $x$ (horizontal, reversed). Outside transformations affect $y$ (vertical, as expected).

Example: $y = -2(x+3)^2 + 5$

Shift left 3 ( $h = -3$ )
Vertical stretch by 2, reflect over $x$ -axis ( $a = -2$ )
Shift up 5 ( $k = 5$ )

Vertex: $(-3, 5)$ , opens downward, narrower than $x^2$ .

📝 Transforming Key Points

Example: Transform $y = x^3$ into $y = -\frac{1}{2}(x-1)^3+4$

$a = -1/2$ , $h = 1$ , $k = 4$ .

Parent $(x, y)$	After shift: $(x+1, y)$	After scale: $(x+1, -y/2+4)$

General Point Transformation

$(x, y) \to \left(\frac{x}{b}+h, \; ay+k\right)$

✏️ Writing Equations from Transformations

Word → Equation

"The graph of $y=\sqrt{x}$ is reflected over the $x$ -axis, stretched vertically by 3, shifted right 2, and shifted down 1."

$y = -3\sqrt{x-2}-1$

Graph → Equation

Identify the parent function
Find the new vertex/key point → determines $h, k$
Check orientation (reflected?) → determines sign of $a$
Use another point to find $|a|$

Example: Parabola, vertex $(2, -1)$ , opens down, passes through $(3, -3)$ .

$y = a(x-2)^2-1$ . $-3 = a(1)^2-1 \implies a = -2$ .

$y = -2(x-2)^2-1$

Combined Transformations Quiz 🎯

Combined Transform Practice 🧮

For $y = -2(x-3)^2+7$ :

1) Vertex $h$ = ?

2) Vertex $k$ = ?

3) The point $(4, y)$ on this graph: $y$ = ?

Order of Transformations 🔽

🔢 Power & Root Functions

Even Powers: $y = x^2, x^4, x^6, ...$

Symmetric about $y$ -axis (even functions)
Shape: U gets flatter near origin, steeper away
Higher power → more "rectangular"

Odd Powers: $y = x^3, x^5, x^7, ...$

Symmetric about origin (odd functions)
Shape: S-curve through origin
Higher power → flatter near 0, steeper far away

Root Functions

$y = x^{1/n}$

Even roots ( $\sqrt{x}, \sqrt[4]{x}$ ): domain

Key Relationship

$y = x^n$ and $y = x^{1/n}$ (same parity) are inverse functions — they reflect across $y = x$ .

⭐ Special Functions

Greatest Integer (Floor) Function

$y = \lfloor x \rfloor$

Step function: jumps at every integer
$\lfloor 2.7 \rfloor = 2$ , $\lfloor -1.3 \rfloor = -2$
Used in pricing (round down), computer science

Piecewise-Defined Functions

$f(x) = \begin{cases} x^2 & x < 0 \\ 2x+1 & x \geq 0 \end{cases}$

Different rules for different intervals
Check continuity at boundary points

Logistic Function

$y = \frac{L}{1 + e^{-k(x-x_0)}}$

S-shaped (sigmoid)
Models population growth, learning curves
Horizontal asymptotes at $y=0$ and $y=L$

Recognizing parent functions is the FIRST step in any transformation problem!

Parent Function Recognition 🎯

Parent Function Properties 🧮

1) $y = x^3$ : $f(-2)$ = ?

2) $y = |x|$ : $f(-5)$ = ?

3) $y = 2^x$ : $f(3)$ = ?

Match Parent Functions 🔽

Graph shows: vertex at $(1, -3)$ , opens up, passes through $(3, 5)$ .

Parent: $y = x^2$
Template: $y = a(x-1)^2-3$
Solve: $5 = a(3-1)^2-3 \implies 8 = 4a \implies a = 2$
Answer: $y = 2(x-1)^2-3$

✏️ Absolute Value & Square Root

Absolute Value: $y = a|x-h|+k$

Vertex at $(h, k)$
Opens up: $a > 0$ ; opens down: $a < 0$
Slope of right branch = $a$ ; left branch = $-a$

Example: V-shape, vertex $(2, 1)$ , passes through $(5, -5)$ .

$-5 = a|5-2|+1 \implies -6 = 3a \implies a = -2$

$y = -2|x-2|+1$

Square Root: $y = a\sqrt{x-h}+k$

Starting point at $(h, k)$ (where the curve begins)
$a > 0$ : curve goes up; $a < 0$ : curve goes down

Example: Starts at $(-1, 3)$ , passes through $(3, 7)$ .

$7 = a\sqrt{3-(-1)}+3 \implies 4 = 2a \implies a = 2$

$y = 2\sqrt{x+1}+3$

🌊 Writing Trig Equations from Graphs

$y = A\sin(B(x-C))+D$ or $y = A\cos(B(x-C))+D$

Parameter	From Graph
$A$ (amplitude)	$\frac{\text{max} - \text{min}}{2}$
$D$ (midline)

Example

Graph: max = 5, min = 1, period = $\pi$ , starts at max when $x = \pi/4$ .

$A = \frac{5-1}{2} = 2$
$D = \frac{5+1}{2} = 3$

$y = 2\cos\!\left(2\!\left(x-\frac{\pi}{4}\right)\right)+3$

Writing Equations Quiz 🎯

Find the Parameter 🧮

1) $y = a(x-1)^2-2$ , passes through $(3, 6)$ . $a$ = ?

2) $y = a\sqrt{x+4}+1$ , passes through $(0, 5)$ . = ?

3) Trig: max = 7, min = 1. Amplitude $A$ = ?

Match Graphs to Equations 🔽

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

Transform point $(x,y) \to \left(\frac{x}{b}+h,\; ay+k\right)$

Even & Odd Functions

Even: $f(-x) = f(x)$ — symmetric about $y$ -axis
Odd: $f(-x) = -f(x)$ — symmetric about origin

🧠 Problem-Solving Strategies

Strategy 1: Transform Key Points

Given $y = -2f(x+1)-3$ and parent points $\{(-2,4), (0,0), (2,4)\}$ :

Parent $(x,y)$	Shift: $(x-1, y)$	Scale: $(x-1, -2y-3)$

Strategy 2: Identify from Description

"Graph $y = \sqrt{x}$ , shift left 4, stretch vertically by 3, reflect, shift up 2."

$y = -3\sqrt{x+4}+2$

Strategy 3: Match Domain/Range

Parent $y = \sqrt{x}$ : domain $[0,\infty)$ , range .

$y = -3\sqrt{x+4}+2$ : domain $[- 4, \infty$ , range .

Domain shifts by $h = -4$ (left 4)
Range: max is $k = 2$ , goes down (reflected)

🔗 Connections to Calculus

Transformations Preserve Shape

If $f'(x_0) = m$ (slope at $x_0$ ), then for $g(x) = af(b(x-h))+k$ :

$g'(x) = ab \cdot f'(b(x-h))$

The derivative scales by $ab$ ! This is the chain rule preview.

Domain & Range Transformations

Operation	Domain	Range
$f(x)+k$	Same	Shifts by $k$
$f(x-h)$

Function Composition as Transformation

$g(x) = 2f(x-3)+1$ is really $g = T \circ f$ where and the input is shifted.

Transformations unite algebra, geometry, and calculus!

Synthesis Quiz 🎯

Master Calculations 🧮

1) $y = 4(x-1)^2-3$ : the $y$ -intercept ( $x=0$ , $y$ = ?)

2) $y = -|x+3|+7$ : the $x$ -intercepts are $x = 4$ and ?

3) Domain of $y = \sqrt{2x-6}$ : $x \geq$ ?

Transformations Master 🔽

(9, 3) \to (9, - 3)

Odd:

f(-x) = -f(x)

→ symmetric about origin (e.g.,

x^3, \sin x

)

Shift up 1:

y = -x^3+1

Odd roots (

\sqrt[3]{x}, \sqrt[5]{x}

): domain all reals

Inverse of corresponding power function

B = \frac{2\pi}{\pi} = 2

Starts at max → use cosine. Phase shift:

C = \pi/4

$(-2, -8)$	$(-1, -8)$	$(-1, 8)$
$(-1, -1)$	$(0, -1)$	$(0, 4.5)$
$(0, 0)$	$(1, 0)$	$(1, 4)$
$(1, 1)$	$(2, 1)$	$(2, 3.5)$
$(2, 8)$	$(3, 8)$	$(3, 0)$

$(-2, 4)$	$(-3, 4)$	$(-3, -11)$
$(0, 0)$	$(-1, 0)$	$(-1, -3)$
$(2, 4)$	$(1, 4)$	$(1, -11)$

Transformations

Transformations - Complete Interactive Lesson

Part 1: Translations

📐 Translations (Shifts)

Vertical Shifts

Horizontal Shifts

Combined Example

📝 Worked Examples

Example 1: y=x+4−3y = \sqrt{x+4} - 3y=x+4​−3

🧠 The General Translation

Part 2: Reflections

🔄 Reflections

Reflection Over the xxx-axis

Reflection Over the yyy-axis

Part 3: Stretches & Compressions

📏 Stretches & Compressions

Vertical Stretch/Compression

Part 4: Combined Transformations

🔗 Combining Multiple Transformations

The General Form

Order of Operations

Part 5: Piecewise Functions

📚 Parent Functions Gallery

The Essential Toolkit

Part 6: Problem-Solving Workshop

📐 Writing Equations from Graphs

The Reverse Problem

Step-by-Step Strategy

Part 7: Review & Applications

🏆 Transformations — Full Synthesis

Master Checklist

Example 2: y=∣x−1∣+2y = |x-1| + 2y=∣x−1∣+2

Why Horizontal Shifts Are "Backwards"

Translating Key Points

Effect on Domain and Range

Quick Reference

📝 Examples

Example 1: y=−xy = -\sqrt{x}y=−x​

Example 2: y=−xy = \sqrt{-x}y=−x​

Even and Odd Functions

🔀 Combining Reflections with Shifts

Example: y=−∣x−2∣+3y = -|x-2|+3y=−∣x−2∣+3

Example: y=(−x)3+1=−x3+1y = (-x)^3 + 1 = -x^3+1y=(−x)3+1=−x3+1

Horizontal Stretch/Compression

📝 Examples

Example 1: y=3sin⁡xy = 3\sin xy=3sinx

Example 2: y=sin⁡(2x)y = \sin(2x)y=sin(2x)

Example 3: y=12x2y = \frac{1}{2}x^2y=21​x2

Key Insight

🔄 Effect on Period & Amplitude

Example: y=4cos⁡(3x)y = 4\cos(3x)y=4cos(3x)

For General Functions

Example: y=−2(x+3)2+5y = -2(x+3)^2 + 5y=−2(x+3)2+5

📝 Transforming Key Points

Example: Transform y=x3y = x^3y=x3 into y=−12(x−1)3+4y = -\frac{1}{2}(x-1)^3+4y=−21​(x−1)3+4

General Point Transformation

✏️ Writing Equations from Transformations

Word → Equation

Graph → Equation

🔢 Power & Root Functions

Even Powers: y=x2,x4,x6,...y = x^2, x^4, x^6, ...y=x2,x4,x6,...

Odd Powers: y=x3,x5,x7,...y = x^3, x^5, x^7, ...y=x3,x5,x7,...

Root Functions

Key Relationship

⭐ Special Functions

Greatest Integer (Floor) Function

Piecewise-Defined Functions

Logistic Function

Example: Parabola

✏️ Absolute Value & Square Root

Absolute Value: y=a∣x−h∣+ky = a|x-h|+ky=a∣x−h∣+k

Square Root: y=ax−h+ky = a\sqrt{x-h}+ky=ax−h​+k

🌊 Writing Trig Equations from Graphs

y=Asin⁡(B(x−C))+Dy = A\sin(B(x-C))+Dy=Asin(B(x−C))+D or y=Acos⁡(B(x−C))+Dy = A\cos(B(x-C))+Dy=Acos(B(x−C))+D

Example

The Master Formula

Even & Odd Functions

🧠 Problem-Solving Strategies

Strategy 1: Transform Key Points

Strategy 2: Identify from Description

Example 1: $y = \sqrt{x+4} - 3$

Reflection Over the $x$ -axis

Reflection Over the $y$ -axis

Example 2: $y = |x-1| + 2$

Example 1: $y = -\sqrt{x}$

Example 2: $y = \sqrt{-x}$

Example: $y = -|x-2|+3$

Example: $y = (-x)^3 + 1 = -x^3+1$

Example 1: $y = 3\sin x$

Example 2: $y = \sin(2x)$

Example 3: $y = \frac{1}{2}x^2$

Example: $y = 4\cos(3x)$

Example: $y = -2(x+3)^2 + 5$

Example: Transform $y = x^3$ into $y = -\frac{1}{2}(x-1)^3+4$

Even Powers: $y = x^2, x^4, x^6, ...$

Odd Powers: $y = x^3, x^5, x^7, ...$

Absolute Value: $y = a|x-h|+k$

Square Root: $y = a\sqrt{x-h}+k$

$y = A\sin(B(x-C))+D$ or $y = A\cos(B(x-C))+D$