Transformations of Functions

📐 Translations (Shifts)

Part 1 of 7

Vertical Shifts

$y = f(x) + k$

Horizontal Shifts

$y = f(x - h)$

⚠️ 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 $2$, up $5$. Vertex: $(2, 5)$.

📝 Worked Examples

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

Start with $y = \sqrt{x}$.

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

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

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

Start with $y = |x|$.

1. Replace $x$ with $x-1$: shift right $1$
2. 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.

🧠 The General Translation

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

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

Translating Key Points

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

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 $[c+k, d+k]$.

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 $x$ = ?

Translation Concepts 🔽

Exit Quiz ✅

🔄 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

$y = f(-x)$

Negate the input: flip the graph left-right.

Quick Reference

📝 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: $(0,0) \to (0,0)$, $(4,2) \to (4,-2)$, $(9,3) \to (9,-3)$.

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., $x^2, \cos x$)
• Odd: $f(-x) = -f(x)$ → symmetric about origin (e.g., $x^3, \sin x$)

💡 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: $y = -|x-2|+3$

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

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

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

1. Start with $y = x^3$
2. Reflect over $y$-axis: $y = (-x)^3 = -x^3$
3. Shift up 1: $y = -x^3+1$

💡 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 🔽

Exit Quiz ✅

📏 Stretches & Compressions

Part 3 of 7

Vertical Stretch/Compression

$y = a \cdot f(x)$

| $|a| > 1$ | Vertical stretch by factor $|a|$ | |:----------|:------------------------------------| | $0 < |a| < 1$ | Vertical compression by factor $|a|$ | | $a < 0$ | Also reflects over $x$-axis |

Horizontal Stretch/Compression

$y = f(bx)$

| $|b| > 1$ | 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

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 🔽

Exit Quiz ✅

🔗 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:

1. Horizontal shift (inside): $x - h$
2. Horizontal stretch/reflect (inside): multiply by $b$
3. Vertical stretch/reflect (outside): multiply by $a$
4. Vertical shift (outside): add $k$

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

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

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

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

📝 Transforming Key Points

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

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

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

1. Identify the parent function
2. Find the new vertex/key point → determines $h, k$
3. Check orientation (reflected?) → determines sign of $a$
4. 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 🔽

Exit Quiz ✅

📚 Parent Functions Gallery

Part 5 of 7

The Essential Toolkit

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

🔢 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 $x \geq 0$
• Odd roots ($\sqrt[3]{x}, \sqrt[5]{x}$): domain all reals
• Inverse of corresponding power function

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 🔽

Exit Quiz ✅

📐 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

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

Example: Parabola

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

Example

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

• $A = \frac{5-1}{2} = 2$
• $D = \frac{5+1}{2} = 3$
• $B = \frac{2\pi}{\pi} = 2$
• Starts at max → use cosine. Phase shift: $C = \pi/4$

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

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 $(0, 5)$. $a$ = ?

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

Match Graphs to Equations 🔽

Exit Quiz ✅

🏆 Transformations — Full Synthesis

Part 7 of 7

Master Checklist

The Master Formula

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

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 $\{(-2,4), (0,0), (2,4)\}$:

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 $[0,\infty)$.

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

• 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

Function Composition as Transformation

$g(x) = 2f(x-3)+1$ is really $g = T \circ f$ where $T(y) = 2y+1$ 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 $x =$ ?

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

Transformations Master 🔽

Exit Quiz ✅