Function Transformations - Ninth Grade Math
Introduction to Function Transformations
Function Transformation: A change to the graph of a function that moves, resizes, or flips it
Parent Function: The basic, simplest form of a function family
Transformed Function: The result after applying transformations to the parent function
Parent Function: The basic, simplest form of a function family
Transformed Function: The result after applying transformations to the parent function
Three Main Types of Transformations:
1. Translations (Shifts): Moving the graph without changing its shape
• Vertical shifts (up/down)
• Horizontal shifts (left/right)
2. Reflections: Flipping the graph over an axis
• Reflection over x-axis
• Reflection over y-axis
3. Dilations (Stretches/Compressions): Changing the size/shape
• Vertical stretch/compression
• Horizontal stretch/compression
1. Translations (Shifts): Moving the graph without changing its shape
• Vertical shifts (up/down)
• Horizontal shifts (left/right)
2. Reflections: Flipping the graph over an axis
• Reflection over x-axis
• Reflection over y-axis
3. Dilations (Stretches/Compressions): Changing the size/shape
• Vertical stretch/compression
• Horizontal stretch/compression
1. Function Transformation Rules
General Transformation Formula:
$$g(x) = a \cdot f(b(x - h)) + k$$
where $f(x)$ is the parent function and:
• $a$: Vertical stretch/compression and reflection
- If $|a| > 1$: vertical stretch
- If $0 < |a| < 1$: vertical compression
- If $a < 0$: reflection over x-axis
• $b$: Horizontal stretch/compression and reflection
- If $|b| > 1$: horizontal compression
- If $0 < |b| < 1$: horizontal stretch
- If $b < 0$: reflection over y-axis
• $h$: Horizontal shift
- Positive $h$: shift RIGHT
- Negative $h$: shift LEFT
• $k$: Vertical shift
- Positive $k$: shift UP
- Negative $k$: shift DOWN
$$g(x) = a \cdot f(b(x - h)) + k$$
where $f(x)$ is the parent function and:
• $a$: Vertical stretch/compression and reflection
- If $|a| > 1$: vertical stretch
- If $0 < |a| < 1$: vertical compression
- If $a < 0$: reflection over x-axis
• $b$: Horizontal stretch/compression and reflection
- If $|b| > 1$: horizontal compression
- If $0 < |b| < 1$: horizontal stretch
- If $b < 0$: reflection over y-axis
• $h$: Horizontal shift
- Positive $h$: shift RIGHT
- Negative $h$: shift LEFT
• $k$: Vertical shift
- Positive $k$: shift UP
- Negative $k$: shift DOWN
Memory Aid - Inside vs Outside:
• INSIDE the function (with x): Affects horizontal (opposite direction!)
• OUTSIDE the function (with f): Affects vertical (same direction)
• INSIDE the function (with x): Affects horizontal (opposite direction!)
• OUTSIDE the function (with f): Affects vertical (same direction)
2. Translations of Functions
Translation: Sliding the entire graph without changing its shape or orientation
Types: Vertical shifts and horizontal shifts
Types: Vertical shifts and horizontal shifts
Vertical Translations (Shifts)
Vertical Shift Formula:
$$g(x) = f(x) + k$$
• If $k > 0$: Shift UP by $k$ units
• If $k < 0$: Shift DOWN by $|k|$ units
Effect: Every y-coordinate increases or decreases by $k$
Formula: $(x, y) \rightarrow (x, y + k)$
$$g(x) = f(x) + k$$
• If $k > 0$: Shift UP by $k$ units
• If $k < 0$: Shift DOWN by $|k|$ units
Effect: Every y-coordinate increases or decreases by $k$
Formula: $(x, y) \rightarrow (x, y + k)$
Example 1: Given $f(x) = x^2$, describe $g(x) = f(x) + 3 = x^2 + 3$
Transformation: Shift UP 3 units
Vertex of $f$: $(0, 0)$
Vertex of $g$: $(0, 3)$
Key points transform:
$f$: $(1, 1)$, $(2, 4)$, $(-1, 1)$
$g$: $(1, 4)$, $(2, 7)$, $(-1, 4)$
Transformation: Shift UP 3 units
Vertex of $f$: $(0, 0)$
Vertex of $g$: $(0, 3)$
Key points transform:
$f$: $(1, 1)$, $(2, 4)$, $(-1, 1)$
$g$: $(1, 4)$, $(2, 7)$, $(-1, 4)$
Example 2: Given $f(x) = |x|$, describe $h(x) = f(x) - 5 = |x| - 5$
Transformation: Shift DOWN 5 units
Vertex moves from $(0, 0)$ to $(0, -5)$
Transformation: Shift DOWN 5 units
Vertex moves from $(0, 0)$ to $(0, -5)$
Horizontal Translations (Shifts)
Horizontal Shift Formula:
$$g(x) = f(x - h)$$
• If $h > 0$: Shift RIGHT by $h$ units
• If $h < 0$: Shift LEFT by $|h|$ units
IMPORTANT: Direction is OPPOSITE of the sign!
• $f(x - 3)$: shift RIGHT 3
• $f(x + 3)$: shift LEFT 3
Effect: Every x-coordinate changes by $h$
Formula: $(x, y) \rightarrow (x + h, y)$
$$g(x) = f(x - h)$$
• If $h > 0$: Shift RIGHT by $h$ units
• If $h < 0$: Shift LEFT by $|h|$ units
IMPORTANT: Direction is OPPOSITE of the sign!
• $f(x - 3)$: shift RIGHT 3
• $f(x + 3)$: shift LEFT 3
Effect: Every x-coordinate changes by $h$
Formula: $(x, y) \rightarrow (x + h, y)$
Example 3: Given $f(x) = x^2$, describe $g(x) = f(x - 2) = (x - 2)^2$
Transformation: Shift RIGHT 2 units
Vertex moves from $(0, 0)$ to $(2, 0)$
Key points:
$f$: $(0, 0)$, $(1, 1)$, $(2, 4)$
$g$: $(2, 0)$, $(3, 1)$, $(4, 4)$
Transformation: Shift RIGHT 2 units
Vertex moves from $(0, 0)$ to $(2, 0)$
Key points:
$f$: $(0, 0)$, $(1, 1)$, $(2, 4)$
$g$: $(2, 0)$, $(3, 1)$, $(4, 4)$
Example 4: Given $f(x) = x^2$, describe $h(x) = f(x + 4) = (x + 4)^2$
Rewrite as: $h(x) = (x - (-4))^2$
Transformation: Shift LEFT 4 units
Vertex moves from $(0, 0)$ to $(-4, 0)$
Rewrite as: $h(x) = (x - (-4))^2$
Transformation: Shift LEFT 4 units
Vertex moves from $(0, 0)$ to $(-4, 0)$
Combined Translations
Combined Shift Formula:
$$g(x) = f(x - h) + k$$
• Horizontal shift by $h$ (opposite of sign)
• Vertical shift by $k$ (same as sign)
• Order doesn't matter - can apply shifts in any order
$$g(x) = f(x - h) + k$$
• Horizontal shift by $h$ (opposite of sign)
• Vertical shift by $k$ (same as sign)
• Order doesn't matter - can apply shifts in any order
Example 5: Given $f(x) = x^2$, describe $g(x) = (x - 3)^2 + 2$
Horizontal: $x - 3$ → shift RIGHT 3 units
Vertical: $+2$ → shift UP 2 units
Vertex moves from $(0, 0)$ to $(3, 2)$
Horizontal: $x - 3$ → shift RIGHT 3 units
Vertical: $+2$ → shift UP 2 units
Vertex moves from $(0, 0)$ to $(3, 2)$
Example 6: Given $f(x) = |x|$, describe $h(x) = |x + 1| - 4$
Horizontal: $x + 1$ → shift LEFT 1 unit
Vertical: $-4$ → shift DOWN 4 units
Vertex moves from $(0, 0)$ to $(-1, -4)$
Horizontal: $x + 1$ → shift LEFT 1 unit
Vertical: $-4$ → shift DOWN 4 units
Vertex moves from $(0, 0)$ to $(-1, -4)$
3. Reflections of Functions
Reflection: Flipping the graph over a line (x-axis or y-axis)
Mirror Image: The reflected graph is a mirror image across the line of reflection
Mirror Image: The reflected graph is a mirror image across the line of reflection
Reflection Over X-Axis
Reflection Over X-Axis:
$$g(x) = -f(x)$$
Effect: Multiply ALL y-values by -1
Coordinate Change: $(x, y) \rightarrow (x, -y)$
Visual: Flips graph upside down
Key Feature: Negative sign is OUTSIDE the function
$$g(x) = -f(x)$$
Effect: Multiply ALL y-values by -1
Coordinate Change: $(x, y) \rightarrow (x, -y)$
Visual: Flips graph upside down
Key Feature: Negative sign is OUTSIDE the function
Example 1: Given $f(x) = x^2$, describe $g(x) = -x^2$
Transformation: Reflection over x-axis
Original opens UP: Parabola with vertex at $(0, 0)$
Reflected opens DOWN: Inverted parabola
Key points:
$f$: $(1, 1)$, $(2, 4)$, $(-1, 1)$
$g$: $(1, -1)$, $(2, -4)$, $(-1, -1)$
Transformation: Reflection over x-axis
Original opens UP: Parabola with vertex at $(0, 0)$
Reflected opens DOWN: Inverted parabola
Key points:
$f$: $(1, 1)$, $(2, 4)$, $(-1, 1)$
$g$: $(1, -1)$, $(2, -4)$, $(-1, -1)$
Example 2: Given $f(x) = |x|$, describe $h(x) = -|x|$
Transformation: Reflection over x-axis
Original: V-shape opening UP
Reflected: ∧-shape opening DOWN
Vertex stays at $(0, 0)$ (on the x-axis)
Transformation: Reflection over x-axis
Original: V-shape opening UP
Reflected: ∧-shape opening DOWN
Vertex stays at $(0, 0)$ (on the x-axis)
Reflection Over Y-Axis
Reflection Over Y-Axis:
$$g(x) = f(-x)$$
Effect: Multiply ALL x-values by -1
Coordinate Change: $(x, y) \rightarrow (-x, y)$
Visual: Flips graph left-to-right
Key Feature: Negative sign is INSIDE the function (with x)
$$g(x) = f(-x)$$
Effect: Multiply ALL x-values by -1
Coordinate Change: $(x, y) \rightarrow (-x, y)$
Visual: Flips graph left-to-right
Key Feature: Negative sign is INSIDE the function (with x)
Example 3: Given $f(x) = 2^x$, describe $g(x) = 2^{-x}$
Transformation: Reflection over y-axis
Key points:
$f$: $(0, 1)$, $(1, 2)$, $(2, 4)$
$g$: $(0, 1)$, $(-1, 2)$, $(-2, 4)$
Original grows to the right, reflected grows to the left
Transformation: Reflection over y-axis
Key points:
$f$: $(0, 1)$, $(1, 2)$, $(2, 4)$
$g$: $(0, 1)$, $(-1, 2)$, $(-2, 4)$
Original grows to the right, reflected grows to the left
Example 4: Given $f(x) = \sqrt{x}$, describe $h(x) = \sqrt{-x}$
Transformation: Reflection over y-axis
Domain change: $[0, \infty)$ becomes $(-\infty, 0]$
Transformation: Reflection over y-axis
Domain change: $[0, \infty)$ becomes $(-\infty, 0]$
Quick Memory Trick:
• Negative OUTSIDE: flip over X-axis (both have "x")
• Negative INSIDE: flip over Y-axis (both have "y")
• Negative OUTSIDE: flip over X-axis (both have "x")
• Negative INSIDE: flip over Y-axis (both have "y")
4. Dilations of Functions
Dilation: Stretching or compressing a graph
Stretch: Makes graph taller/wider
Compression: Makes graph shorter/narrower
Stretch: Makes graph taller/wider
Compression: Makes graph shorter/narrower
Vertical Dilations
Vertical Stretch/Compression:
$$g(x) = a \cdot f(x)$$ where $a > 0$
If $a > 1$: VERTICAL STRETCH (taller, narrower)
• Multiply all y-values by $a$
• Graph pulled away from x-axis
If $0 < a < 1$: VERTICAL COMPRESSION (shorter, wider)
• Multiply all y-values by $a$ (fraction)
• Graph pushed toward x-axis
Coordinate Change: $(x, y) \rightarrow (x, ay)$
$$g(x) = a \cdot f(x)$$ where $a > 0$
If $a > 1$: VERTICAL STRETCH (taller, narrower)
• Multiply all y-values by $a$
• Graph pulled away from x-axis
If $0 < a < 1$: VERTICAL COMPRESSION (shorter, wider)
• Multiply all y-values by $a$ (fraction)
• Graph pushed toward x-axis
Coordinate Change: $(x, y) \rightarrow (x, ay)$
Example 1: Given $f(x) = x^2$, describe $g(x) = 3x^2$
Transformation: Vertical stretch by factor of 3
Effect: Parabola becomes narrower/steeper
Key points:
$f$: $(1, 1)$, $(2, 4)$
$g$: $(1, 3)$, $(2, 12)$ (y-values tripled)
Transformation: Vertical stretch by factor of 3
Effect: Parabola becomes narrower/steeper
Key points:
$f$: $(1, 1)$, $(2, 4)$
$g$: $(1, 3)$, $(2, 12)$ (y-values tripled)
Example 2: Given $f(x) = x^2$, describe $h(x) = \frac{1}{2}x^2$
Transformation: Vertical compression by factor of $\frac{1}{2}$
Effect: Parabola becomes wider/less steep
Key points:
$f$: $(2, 4)$, $(4, 16)$
$h$: $(2, 2)$, $(4, 8)$ (y-values halved)
Transformation: Vertical compression by factor of $\frac{1}{2}$
Effect: Parabola becomes wider/less steep
Key points:
$f$: $(2, 4)$, $(4, 16)$
$h$: $(2, 2)$, $(4, 8)$ (y-values halved)
Horizontal Dilations
Horizontal Stretch/Compression:
$$g(x) = f(bx)$$ where $b > 0$
If $b > 1$: HORIZONTAL COMPRESSION (narrower)
• Divide all x-values by $b$
• Graph pushed toward y-axis
If $0 < b < 1$: HORIZONTAL STRETCH (wider)
• Divide all x-values by $b$ (multiply by $\frac{1}{b}$)
• Graph pulled away from y-axis
Coordinate Change: $(x, y) \rightarrow \left(\frac{x}{b}, y\right)$
IMPORTANT: Effect is OPPOSITE of what you expect!
$$g(x) = f(bx)$$ where $b > 0$
If $b > 1$: HORIZONTAL COMPRESSION (narrower)
• Divide all x-values by $b$
• Graph pushed toward y-axis
If $0 < b < 1$: HORIZONTAL STRETCH (wider)
• Divide all x-values by $b$ (multiply by $\frac{1}{b}$)
• Graph pulled away from y-axis
Coordinate Change: $(x, y) \rightarrow \left(\frac{x}{b}, y\right)$
IMPORTANT: Effect is OPPOSITE of what you expect!
Example 3: Given $f(x) = x^2$, describe $g(x) = (2x)^2$
Transformation: Horizontal compression by factor of $\frac{1}{2}$
Effect: Parabola becomes narrower
Key points:
$f$: $(2, 4)$, $(4, 16)$
$g$: $(1, 4)$, $(2, 16)$ (x-values halved)
Transformation: Horizontal compression by factor of $\frac{1}{2}$
Effect: Parabola becomes narrower
Key points:
$f$: $(2, 4)$, $(4, 16)$
$g$: $(1, 4)$, $(2, 16)$ (x-values halved)
Example 4: Given $f(x) = |x|$, describe $h(x) = \left|\frac{1}{2}x\right|$
Transformation: Horizontal stretch by factor of 2
Effect: V-shape becomes wider
Key points:
$f$: $(2, 2)$, $(4, 4)$
$h$: $(4, 2)$, $(8, 4)$ (x-values doubled)
Transformation: Horizontal stretch by factor of 2
Effect: V-shape becomes wider
Key points:
$f$: $(2, 2)$, $(4, 4)$
$h$: $(4, 2)$, $(8, 4)$ (x-values doubled)
Dilation Summary:
• Vertical dilation: coefficient OUTSIDE, acts as expected
• Horizontal dilation: coefficient INSIDE, acts OPPOSITE
• Greater than 1: vertical stretch OR horizontal compression
• Between 0 and 1: vertical compression OR horizontal stretch
• Vertical dilation: coefficient OUTSIDE, acts as expected
• Horizontal dilation: coefficient INSIDE, acts OPPOSITE
• Greater than 1: vertical stretch OR horizontal compression
• Between 0 and 1: vertical compression OR horizontal stretch
5. Transformations of Functions (Combined)
General Transformation Formula (Complete):
$$g(x) = a \cdot f(b(x - h)) + k$$
Order of Operations (from inside out):
1. Horizontal shift by $h$
2. Horizontal stretch/compression by $\frac{1}{b}$
3. Reflection over y-axis if $b < 0$
4. Vertical stretch/compression by $|a|$
5. Reflection over x-axis if $a < 0$
6. Vertical shift by $k$
$$g(x) = a \cdot f(b(x - h)) + k$$
Order of Operations (from inside out):
1. Horizontal shift by $h$
2. Horizontal stretch/compression by $\frac{1}{b}$
3. Reflection over y-axis if $b < 0$
4. Vertical stretch/compression by $|a|$
5. Reflection over x-axis if $a < 0$
6. Vertical shift by $k$
Example 1: Describe all transformations: $g(x) = 2(x - 3)^2 + 5$
Parent function: $f(x) = x^2$
Transformations:
• $a = 2$: Vertical stretch by 2
• $h = 3$: Shift RIGHT 3 units
• $k = 5$: Shift UP 5 units
Vertex moves from $(0, 0)$ to $(3, 5)$
Parabola is narrower and higher
Parent function: $f(x) = x^2$
Transformations:
• $a = 2$: Vertical stretch by 2
• $h = 3$: Shift RIGHT 3 units
• $k = 5$: Shift UP 5 units
Vertex moves from $(0, 0)$ to $(3, 5)$
Parabola is narrower and higher
Example 2: Describe: $h(x) = -\frac{1}{2}|x + 4| - 1$
Parent: $f(x) = |x|$
Transformations:
• $a = -\frac{1}{2}$: Reflection over x-axis AND vertical compression by $\frac{1}{2}$
• $h = -4$: Shift LEFT 4 units
• $k = -1$: Shift DOWN 1 unit
Result: Upside-down V, wider, moved to $(-4, -1)$
Parent: $f(x) = |x|$
Transformations:
• $a = -\frac{1}{2}$: Reflection over x-axis AND vertical compression by $\frac{1}{2}$
• $h = -4$: Shift LEFT 4 units
• $k = -1$: Shift DOWN 1 unit
Result: Upside-down V, wider, moved to $(-4, -1)$
Example 3: Describe: $g(x) = -3(x - 1)^2 + 7$
Transformations:
• Vertical stretch by 3
• Reflection over x-axis (opens down)
• Shift RIGHT 1 unit
• Shift UP 7 units
Vertex: $(1, 7)$ with parabola opening downward
Transformations:
• Vertical stretch by 3
• Reflection over x-axis (opens down)
• Shift RIGHT 1 unit
• Shift UP 7 units
Vertex: $(1, 7)$ with parabola opening downward
6. Describe Function Transformations
Steps to Describe Transformations:
Step 1: Identify the parent function
Step 2: Look for coefficient outside (vertical dilation/reflection)
Step 3: Look inside parentheses for horizontal changes
Step 4: Look for constant outside (vertical shift)
Step 5: List all transformations in order
Step 6: Describe the overall effect
Step 1: Identify the parent function
Step 2: Look for coefficient outside (vertical dilation/reflection)
Step 3: Look inside parentheses for horizontal changes
Step 4: Look for constant outside (vertical shift)
Step 5: List all transformations in order
Step 6: Describe the overall effect
Example 1: Describe the transformation from $f(x) = x^2$ to $g(x) = (x + 2)^2 - 3$
Analysis:
• $(x + 2)$: Horizontal shift LEFT 2 units
• $-3$: Vertical shift DOWN 3 units
Description: The graph of $f$ is translated 2 units left and 3 units down to produce $g$. The vertex moves from $(0, 0)$ to $(-2, -3)$.
Analysis:
• $(x + 2)$: Horizontal shift LEFT 2 units
• $-3$: Vertical shift DOWN 3 units
Description: The graph of $f$ is translated 2 units left and 3 units down to produce $g$. The vertex moves from $(0, 0)$ to $(-2, -3)$.
Example 2: Describe: From $f(x) = |x|$ to $g(x) = -2|x - 1| + 4$
Transformations in order:
1. Shift RIGHT 1 unit: $(x - 1)$
2. Vertical stretch by factor of 2: coefficient 2
3. Reflection over x-axis: negative sign
4. Shift UP 4 units: $+4$
Description: The graph is shifted right 1 unit, stretched vertically by 2, flipped upside down, and shifted up 4 units. The vertex moves from $(0, 0)$ to $(1, 4)$ and the V now opens downward.
Transformations in order:
1. Shift RIGHT 1 unit: $(x - 1)$
2. Vertical stretch by factor of 2: coefficient 2
3. Reflection over x-axis: negative sign
4. Shift UP 4 units: $+4$
Description: The graph is shifted right 1 unit, stretched vertically by 2, flipped upside down, and shifted up 4 units. The vertex moves from $(0, 0)$ to $(1, 4)$ and the V now opens downward.
Example 3: Write the equation after transforming $f(x) = x^2$ as follows:
• Shift right 5 units
• Shift down 2 units
• Vertical stretch by 3
Solution:
Start with $f(x) = x^2$
Right 5: $f(x - 5) = (x - 5)^2$
Stretch by 3: $3(x - 5)^2$
Down 2: $3(x - 5)^2 - 2$
Answer: $g(x) = 3(x - 5)^2 - 2$
• Shift right 5 units
• Shift down 2 units
• Vertical stretch by 3
Solution:
Start with $f(x) = x^2$
Right 5: $f(x - 5) = (x - 5)^2$
Stretch by 3: $3(x - 5)^2$
Down 2: $3(x - 5)^2 - 2$
Answer: $g(x) = 3(x - 5)^2 - 2$
Complete Transformation Reference Table
| Transformation | Notation | Effect | Coordinate Change |
|---|---|---|---|
| Vertical Shift UP | $f(x) + k$, $k > 0$ | Move graph up $k$ units | $(x, y) \rightarrow (x, y+k)$ |
| Vertical Shift DOWN | $f(x) - k$, $k > 0$ | Move graph down $k$ units | $(x, y) \rightarrow (x, y-k)$ |
| Horizontal Shift RIGHT | $f(x - h)$, $h > 0$ | Move graph right $h$ units | $(x, y) \rightarrow (x+h, y)$ |
| Horizontal Shift LEFT | $f(x + h)$, $h > 0$ | Move graph left $h$ units | $(x, y) \rightarrow (x-h, y)$ |
| Reflection over X-axis | $-f(x)$ | Flip over x-axis | $(x, y) \rightarrow (x, -y)$ |
| Reflection over Y-axis | $f(-x)$ | Flip over y-axis | $(x, y) \rightarrow (-x, y)$ |
| Vertical Stretch | $a \cdot f(x)$, $a > 1$ | Stretch away from x-axis | $(x, y) \rightarrow (x, ay)$ |
| Vertical Compression | $a \cdot f(x)$, $0 < a < 1$ | Compress toward x-axis | $(x, y) \rightarrow (x, ay)$ |
| Horizontal Compression | $f(bx)$, $b > 1$ | Compress toward y-axis | $(x, y) \rightarrow (x/b, y)$ |
| Horizontal Stretch | $f(bx)$, $0 < b < 1$ | Stretch away from y-axis | $(x, y) \rightarrow (x/b, y)$ |
Quick Decision Guide
| Where is the change? | Type | Direction/Effect |
|---|---|---|
| Added/Subtracted OUTSIDE | Vertical shift | + means UP, - means DOWN |
| Added/Subtracted INSIDE | Horizontal shift | - means RIGHT, + means LEFT (opposite!) |
| Multiplied OUTSIDE | Vertical dilation/reflection | $|a| > 1$: stretch; $0 < |a| < 1$: compress; $a < 0$: flip |
| Multiplied INSIDE | Horizontal dilation/reflection | $|b| > 1$: compress; $0 < |b| < 1$: stretch; $b < 0$: flip |
Common Parent Functions
| Name | Parent Function | Basic Shape |
|---|---|---|
| Linear | $f(x) = x$ | Straight line through origin, slope 1 |
| Quadratic | $f(x) = x^2$ | Parabola, vertex at origin, opens up |
| Cubic | $f(x) = x^3$ | S-curve through origin |
| Absolute Value | $f(x) = |x|$ | V-shape, vertex at origin |
| Square Root | $f(x) = \sqrt{x}$ | Half-parabola, starts at origin, goes right |
| Exponential | $f(x) = 2^x$ or $e^x$ | Curved growth, passes through $(0, 1)$ |
Success Tips for Function Transformations:
✓ INSIDE changes (with x) affect horizontal → OPPOSITE direction
✓ OUTSIDE changes (with f) affect vertical → SAME direction
✓ Start with parent function, apply transformations step by step
✓ Horizontal shifts: $f(x-h)$ goes RIGHT, $f(x+h)$ goes LEFT
✓ Vertical shifts: $f(x)+k$ goes UP, $f(x)-k$ goes DOWN
✓ Negative outside → flip over x-axis
✓ Negative inside → flip over y-axis
✓ Vertical stretch: $|a| > 1$; compression: $0 < |a| < 1$
✓ Horizontal is opposite: $|b| > 1$ compresses, $0 < |b| < 1$ stretches
✓ Practice by tracking key points through transformations!
✓ INSIDE changes (with x) affect horizontal → OPPOSITE direction
✓ OUTSIDE changes (with f) affect vertical → SAME direction
✓ Start with parent function, apply transformations step by step
✓ Horizontal shifts: $f(x-h)$ goes RIGHT, $f(x+h)$ goes LEFT
✓ Vertical shifts: $f(x)+k$ goes UP, $f(x)-k$ goes DOWN
✓ Negative outside → flip over x-axis
✓ Negative inside → flip over y-axis
✓ Vertical stretch: $|a| > 1$; compression: $0 < |a| < 1$
✓ Horizontal is opposite: $|b| > 1$ compresses, $0 < |b| < 1$ stretches
✓ Practice by tracking key points through transformations!