Absolute Value Functions - Ninth Grade Math
Introduction to Absolute Value
Absolute Value: The distance of a number from zero on a number line
Key Property: Absolute value is ALWAYS non-negative (≥ 0)
Symbol: Two vertical bars around the number or expression: $|x|$
Meaning: "How far from zero?" regardless of direction
Key Property: Absolute value is ALWAYS non-negative (≥ 0)
Symbol: Two vertical bars around the number or expression: $|x|$
Meaning: "How far from zero?" regardless of direction
Absolute Value Definition:
$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
In Words:
• If the number is positive or zero, absolute value equals the number itself
• If the number is negative, absolute value equals its opposite (makes it positive)
$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
In Words:
• If the number is positive or zero, absolute value equals the number itself
• If the number is negative, absolute value equals its opposite (makes it positive)
Basic Examples:
$|5| = 5$ (already positive)
$|-5| = 5$ (opposite of -5)
$|0| = 0$ (zero stays zero)
$|-12| = 12$
$|7.3| = 7.3$
$|-\frac{1}{2}| = \frac{1}{2}$
$|5| = 5$ (already positive)
$|-5| = 5$ (opposite of -5)
$|0| = 0$ (zero stays zero)
$|-12| = 12$
$|7.3| = 7.3$
$|-\frac{1}{2}| = \frac{1}{2}$
Think of it this way:
Absolute value "removes" the negative sign if there is one!
Absolute value "removes" the negative sign if there is one!
1. Complete Function Tables: Absolute Value Functions
Absolute Value Function: A function where the variable appears inside absolute value bars
Parent Function: $f(x) = |x|$
Parent Function: $f(x) = |x|$
Steps to Complete Table:
Step 1: Substitute the x-value into the function
Step 2: Evaluate what's inside the absolute value bars first
Step 3: Take the absolute value
Step 4: Perform any operations outside the bars
Step 1: Substitute the x-value into the function
Step 2: Evaluate what's inside the absolute value bars first
Step 3: Take the absolute value
Step 4: Perform any operations outside the bars
Example 1: Complete table for $f(x) = |x|$
Calculate:
$f(-3) = |-3| = 3$
$f(-2) = |-2| = 2$
$f(-1) = |-1| = 1$
$f(0) = |0| = 0$
$f(1) = |1| = 1$
$f(2) = |2| = 2$
$f(3) = |3| = 3$
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| f(x) | ? | ? | ? | ? | ? | ? | ? |
Calculate:
$f(-3) = |-3| = 3$
$f(-2) = |-2| = 2$
$f(-1) = |-1| = 1$
$f(0) = |0| = 0$
$f(1) = |1| = 1$
$f(2) = |2| = 2$
$f(3) = |3| = 3$
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| f(x) | 3 | 2 | 1 | 0 | 1 | 2 | 3 |
Example 2: Complete table for $g(x) = |x - 2|$
Calculate (evaluate inside first!):
$g(-1) = |-1 - 2| = |-3| = 3$
$g(0) = |0 - 2| = |-2| = 2$
$g(1) = |1 - 2| = |-1| = 1$
$g(2) = |2 - 2| = |0| = 0$
$g(3) = |3 - 2| = |1| = 1$
$g(4) = |4 - 2| = |2| = 2$
| x | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| g(x) | ? | ? | ? | ? | ? | ? |
Calculate (evaluate inside first!):
$g(-1) = |-1 - 2| = |-3| = 3$
$g(0) = |0 - 2| = |-2| = 2$
$g(1) = |1 - 2| = |-1| = 1$
$g(2) = |2 - 2| = |0| = 0$
$g(3) = |3 - 2| = |1| = 1$
$g(4) = |4 - 2| = |2| = 2$
| x | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| g(x) | 3 | 2 | 1 | 0 | 1 | 2 |
Example 3: Complete table for $h(x) = 2|x| + 1$
For $x = -2$:
$h(-2) = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5$
For $x = 0$:
$h(0) = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1$
For $x = 3$:
$h(3) = 2|3| + 1 = 2(3) + 1 = 6 + 1 = 7$
For $x = -2$:
$h(-2) = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5$
For $x = 0$:
$h(0) = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1$
For $x = 3$:
$h(3) = 2|3| + 1 = 2(3) + 1 = 6 + 1 = 7$
2. Graph an Absolute Value Function
Graph Shape: V-shape (like the letter V)
Vertex: The point at the bottom (or top) of the V
Parent Function: $f(x) = |x|$ has vertex at origin $(0, 0)$
Vertex: The point at the bottom (or top) of the V
Parent Function: $f(x) = |x|$ has vertex at origin $(0, 0)$
General Form of Absolute Value Function:
$$f(x) = a|x - h| + k$$
where:
• $a$ = vertical stretch/compression and reflection
• $h$ = horizontal shift (vertex x-coordinate)
• $k$ = vertical shift (vertex y-coordinate)
• Vertex: $(h, k)$
$$f(x) = a|x - h| + k$$
where:
• $a$ = vertical stretch/compression and reflection
• $h$ = horizontal shift (vertex x-coordinate)
• $k$ = vertical shift (vertex y-coordinate)
• Vertex: $(h, k)$
Parent Function: $f(x) = |x|$
Key Features of $f(x) = |x|$:
• Vertex: $(0, 0)$
• Opens: Upward
• Slope: -1 on left side, +1 on right side
• Shape: V pointing up
• Axis of Symmetry: $x = 0$ (y-axis)
• Vertex: $(0, 0)$
• Opens: Upward
• Slope: -1 on left side, +1 on right side
• Shape: V pointing up
• Axis of Symmetry: $x = 0$ (y-axis)
Example 1: Graph $f(x) = |x|$
Key Points:
Description:
• Vertex at origin $(0, 0)$
• V-shape opening upward
• Symmetric about y-axis
• Points: $(-2, 2)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, $(2, 2)$
Key Points:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 3 | 2 | 1 | 0 | 1 | 2 | 3 |
Description:
• Vertex at origin $(0, 0)$
• V-shape opening upward
• Symmetric about y-axis
• Points: $(-2, 2)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, $(2, 2)$
Steps to Graph Absolute Value Function:
Step 1: Identify vertex $(h, k)$ from $f(x) = a|x - h| + k$
Step 2: Plot the vertex
Step 3: Determine if it opens up ($a > 0$) or down ($a < 0$)
Step 4: Find additional points on both sides of vertex
Step 5: Connect with V-shape
Step 6: Draw arrows at ends
Step 1: Identify vertex $(h, k)$ from $f(x) = a|x - h| + k$
Step 2: Plot the vertex
Step 3: Determine if it opens up ($a > 0$) or down ($a < 0$)
Step 4: Find additional points on both sides of vertex
Step 5: Connect with V-shape
Step 6: Draw arrows at ends
Example 2: Graph $g(x) = |x - 2| + 1$
Vertex: $(h, k) = (2, 1)$
Opens: Upward ($a = 1 > 0$)
Additional points:
$g(0) = |0 - 2| + 1 = 2 + 1 = 3$ → $(0, 3)$
$g(1) = |1 - 2| + 1 = 1 + 1 = 2$ → $(1, 2)$
$g(2) = |2 - 2| + 1 = 0 + 1 = 1$ → $(2, 1)$ (vertex)
$g(3) = |3 - 2| + 1 = 1 + 1 = 2$ → $(3, 2)$
$g(4) = |4 - 2| + 1 = 2 + 1 = 3$ → $(4, 3)$
V-shape with vertex at $(2, 1)$
Vertex: $(h, k) = (2, 1)$
Opens: Upward ($a = 1 > 0$)
Additional points:
$g(0) = |0 - 2| + 1 = 2 + 1 = 3$ → $(0, 3)$
$g(1) = |1 - 2| + 1 = 1 + 1 = 2$ → $(1, 2)$
$g(2) = |2 - 2| + 1 = 0 + 1 = 1$ → $(2, 1)$ (vertex)
$g(3) = |3 - 2| + 1 = 1 + 1 = 2$ → $(3, 2)$
$g(4) = |4 - 2| + 1 = 2 + 1 = 3$ → $(4, 3)$
V-shape with vertex at $(2, 1)$
Example 3: Graph $h(x) = -|x| + 3$
Vertex: $(0, 3)$
Opens: Downward ($a = -1 < 0$)
Shape: Upside-down V (∧-shape)
Points:
$(-2, 1)$, $(-1, 2)$, $(0, 3)$, $(1, 2)$, $(2, 1)$
Vertex: $(0, 3)$
Opens: Downward ($a = -1 < 0$)
Shape: Upside-down V (∧-shape)
Points:
$(-2, 1)$, $(-1, 2)$, $(0, 3)$, $(1, 2)$, $(2, 1)$
3-4. Domain and Range of Absolute Value Functions
Domain: All possible input values (x-values)
Range: All possible output values (y-values)
Range: All possible output values (y-values)
For Absolute Value Functions:
Domain: ALWAYS all real numbers
$$\text{Domain: } (-\infty, \infty) \text{ or } \{x | x \in \mathbb{R}\}$$
Range: Depends on vertex and direction
For $f(x) = a|x - h| + k$:
• If $a > 0$ (opens UP):
Minimum value at vertex: $y = k$
$$\text{Range: } [k, \infty) \text{ or } \{y | y \geq k\}$$
• If $a < 0$ (opens DOWN):
Maximum value at vertex: $y = k$
$$\text{Range: } (-\infty, k] \text{ or } \{y | y \leq k\}$$
Domain: ALWAYS all real numbers
$$\text{Domain: } (-\infty, \infty) \text{ or } \{x | x \in \mathbb{R}\}$$
Range: Depends on vertex and direction
For $f(x) = a|x - h| + k$:
• If $a > 0$ (opens UP):
Minimum value at vertex: $y = k$
$$\text{Range: } [k, \infty) \text{ or } \{y | y \geq k\}$$
• If $a < 0$ (opens DOWN):
Maximum value at vertex: $y = k$
$$\text{Range: } (-\infty, k] \text{ or } \{y | y \leq k\}$$
Example 1: Find domain and range of $f(x) = |x|$
Domain: Can input any real number → $(-\infty, \infty)$
Vertex: $(0, 0)$
Opens: Upward
Minimum value: $y = 0$
Range: $[0, \infty)$ or $y \geq 0$
Domain: Can input any real number → $(-\infty, \infty)$
Vertex: $(0, 0)$
Opens: Upward
Minimum value: $y = 0$
Range: $[0, \infty)$ or $y \geq 0$
Example 2: Find domain and range of $g(x) = |x - 3| + 2$
Domain: $(-\infty, \infty)$
Vertex: $(3, 2)$
Opens: Upward ($a = 1 > 0$)
Minimum value: $y = 2$
Range: $[2, \infty)$ or $y \geq 2$
Domain: $(-\infty, \infty)$
Vertex: $(3, 2)$
Opens: Upward ($a = 1 > 0$)
Minimum value: $y = 2$
Range: $[2, \infty)$ or $y \geq 2$
Example 3: Find domain and range of $h(x) = -|x + 1| + 5$
Domain: $(-\infty, \infty)$
Vertex: $(-1, 5)$
Opens: Downward ($a = -1 < 0$)
Maximum value: $y = 5$
Range: $(-\infty, 5]$ or $y \leq 5$
Domain: $(-\infty, \infty)$
Vertex: $(-1, 5)$
Opens: Downward ($a = -1 < 0$)
Maximum value: $y = 5$
Range: $(-\infty, 5]$ or $y \leq 5$
Example 4: Find domain and range of $f(x) = 2|x| - 3$
Domain: $(-\infty, \infty)$
Vertex: $(0, -3)$
Opens: Upward ($a = 2 > 0$)
Minimum: $y = -3$
Range: $[-3, \infty)$ or $y \geq -3$
Domain: $(-\infty, \infty)$
Vertex: $(0, -3)$
Opens: Upward ($a = 2 > 0$)
Minimum: $y = -3$
Range: $[-3, \infty)$ or $y \geq -3$
Quick Tips:
• Domain is ALWAYS all real numbers
• Find vertex to determine range
• Opens up → range starts at $k$, goes to $\infty$
• Opens down → range goes from $-\infty$ to $k$
• Domain is ALWAYS all real numbers
• Find vertex to determine range
• Opens up → range starts at $k$, goes to $\infty$
• Opens down → range goes from $-\infty$ to $k$
5-6. Transformations of Absolute Value Functions
Transformation: Changes to the parent function $f(x) = |x|$
Types: Translations (shifts), Reflections, and Dilations (stretches/compressions)
Types: Translations (shifts), Reflections, and Dilations (stretches/compressions)
Transformation Formula:
$$f(x) = a|x - h| + k$$
Each parameter causes a transformation:
$a$ (Vertical Stretch/Compression & Reflection):
• If $|a| > 1$: Vertical stretch (narrower V)
• If $0 < |a| < 1$: Vertical compression (wider V)
• If $a < 0$: Reflection over x-axis (flips upside down)
$h$ (Horizontal Translation):
• $|x - h|$: Shift RIGHT by $h$ units
• $|x + h|$: Shift LEFT by $h$ units
NOTE: Opposite of sign!
$k$ (Vertical Translation):
• $+ k$: Shift UP by $k$ units
• $- k$: Shift DOWN by $k$ units
$$f(x) = a|x - h| + k$$
Each parameter causes a transformation:
$a$ (Vertical Stretch/Compression & Reflection):
• If $|a| > 1$: Vertical stretch (narrower V)
• If $0 < |a| < 1$: Vertical compression (wider V)
• If $a < 0$: Reflection over x-axis (flips upside down)
$h$ (Horizontal Translation):
• $|x - h|$: Shift RIGHT by $h$ units
• $|x + h|$: Shift LEFT by $h$ units
NOTE: Opposite of sign!
$k$ (Vertical Translation):
• $+ k$: Shift UP by $k$ units
• $- k$: Shift DOWN by $k$ units
Type 1: Vertical Translations (Up/Down)
Vertical Shift Formula: $f(x) = |x| + k$
• $f(x) = |x| + k$: Shifts UP by $k$ units
• $f(x) = |x| - k$: Shifts DOWN by $k$ units
• Vertex moves from $(0, 0)$ to $(0, k)$
• $f(x) = |x| + k$: Shifts UP by $k$ units
• $f(x) = |x| - k$: Shifts DOWN by $k$ units
• Vertex moves from $(0, 0)$ to $(0, k)$
Example 1: Describe transformation: $f(x) = |x| + 3$
Parent: $f(x) = |x|$ with vertex at $(0, 0)$
Transformation: Shift UP 3 units
New Vertex: $(0, 3)$
All points move up 3 units
Parent: $f(x) = |x|$ with vertex at $(0, 0)$
Transformation: Shift UP 3 units
New Vertex: $(0, 3)$
All points move up 3 units
Example 2: Describe: $g(x) = |x| - 5$
Transformation: Shift DOWN 5 units
Vertex: $(0, -5)$
Transformation: Shift DOWN 5 units
Vertex: $(0, -5)$
Type 2: Horizontal Translations (Left/Right)
Horizontal Shift Formula: $f(x) = |x - h|$
• $f(x) = |x - h|$: Shifts RIGHT by $h$ units
• $f(x) = |x + h|$: Shifts LEFT by $h$ units
• OPPOSITE of sign inside bars!
• Vertex moves from $(0, 0)$ to $(h, 0)$
• $f(x) = |x - h|$: Shifts RIGHT by $h$ units
• $f(x) = |x + h|$: Shifts LEFT by $h$ units
• OPPOSITE of sign inside bars!
• Vertex moves from $(0, 0)$ to $(h, 0)$
Example 3: Describe: $f(x) = |x - 4|$
Transformation: Shift RIGHT 4 units
Vertex: $(4, 0)$
Remember: $(x - 4)$ means shift right!
Transformation: Shift RIGHT 4 units
Vertex: $(4, 0)$
Remember: $(x - 4)$ means shift right!
Example 4: Describe: $g(x) = |x + 2|$
Rewrite as: $g(x) = |x - (-2)|$
Transformation: Shift LEFT 2 units
Vertex: $(-2, 0)$
Rewrite as: $g(x) = |x - (-2)|$
Transformation: Shift LEFT 2 units
Vertex: $(-2, 0)$
Type 3: Combined Translations
Example 5: Describe: $f(x) = |x - 3| + 2$
Horizontal: Shift RIGHT 3 units
Vertical: Shift UP 2 units
Vertex: $(3, 2)$
Opens: Upward
Horizontal: Shift RIGHT 3 units
Vertical: Shift UP 2 units
Vertex: $(3, 2)$
Opens: Upward
Example 6: Describe: $g(x) = |x + 1| - 4$
Horizontal: Shift LEFT 1 unit
Vertical: Shift DOWN 4 units
Vertex: $(-1, -4)$
Horizontal: Shift LEFT 1 unit
Vertical: Shift DOWN 4 units
Vertex: $(-1, -4)$
Type 4: Reflections
Reflection Formula: $f(x) = -|x|$
• Negative sign in front: Flips graph over x-axis
• V-shape becomes upside-down V (∧-shape)
• Opens downward instead of upward
• Negative sign in front: Flips graph over x-axis
• V-shape becomes upside-down V (∧-shape)
• Opens downward instead of upward
Example 7: Describe: $f(x) = -|x|$
Transformation: Reflection over x-axis
Vertex: $(0, 0)$ (unchanged)
Opens: Downward (∧-shape)
Points: $(-2, -2)$, $(-1, -1)$, $(0, 0)$, $(1, -1)$, $(2, -2)$
Transformation: Reflection over x-axis
Vertex: $(0, 0)$ (unchanged)
Opens: Downward (∧-shape)
Points: $(-2, -2)$, $(-1, -1)$, $(0, 0)$, $(1, -1)$, $(2, -2)$
Example 8: Describe: $g(x) = -|x - 1| + 3$
Reflection: Over x-axis (opens down)
Horizontal: Right 1 unit
Vertical: Up 3 units
Vertex: $(1, 3)$
Opens: Downward
Reflection: Over x-axis (opens down)
Horizontal: Right 1 unit
Vertical: Up 3 units
Vertex: $(1, 3)$
Opens: Downward
Type 5: Vertical Stretches and Compressions (Dilations)
Vertical Dilation Formula: $f(x) = a|x|$ where $a \neq 1$
• If $|a| > 1$: Vertical STRETCH (narrower, steeper)
• If $0 < |a| < 1$: Vertical COMPRESSION (wider, less steep)
• Multiplies all y-values by $a$
• If $|a| > 1$: Vertical STRETCH (narrower, steeper)
• If $0 < |a| < 1$: Vertical COMPRESSION (wider, less steep)
• Multiplies all y-values by $a$
Example 9: Describe: $f(x) = 2|x|$
Transformation: Vertical stretch by factor of 2
Effect: V is narrower/steeper
Vertex: $(0, 0)$
Comparison:
• Original: $(1, 1)$, $(2, 2)$
• Stretched: $(1, 2)$, $(2, 4)$ (y-values doubled)
Transformation: Vertical stretch by factor of 2
Effect: V is narrower/steeper
Vertex: $(0, 0)$
Comparison:
• Original: $(1, 1)$, $(2, 2)$
• Stretched: $(1, 2)$, $(2, 4)$ (y-values doubled)
Example 10: Describe: $g(x) = \frac{1}{2}|x|$
Transformation: Vertical compression by $\frac{1}{2}$
Effect: V is wider/less steep
Comparison:
• Original: $(2, 2)$, $(4, 4)$
• Compressed: $(2, 1)$, $(4, 2)$ (y-values halved)
Transformation: Vertical compression by $\frac{1}{2}$
Effect: V is wider/less steep
Comparison:
• Original: $(2, 2)$, $(4, 4)$
• Compressed: $(2, 1)$, $(4, 2)$ (y-values halved)
Type 6: Combined Transformations
Example 11: Describe all transformations: $f(x) = -3|x + 2| - 1$
$a = -3$:
• Reflection over x-axis (negative sign)
• Vertical stretch by 3 (makes it steeper)
$h = -2$: Shift LEFT 2 units
$k = -1$: Shift DOWN 1 unit
Vertex: $(-2, -1)$
Opens: Downward
Shape: Steep upside-down V
$a = -3$:
• Reflection over x-axis (negative sign)
• Vertical stretch by 3 (makes it steeper)
$h = -2$: Shift LEFT 2 units
$k = -1$: Shift DOWN 1 unit
Vertex: $(-2, -1)$
Opens: Downward
Shape: Steep upside-down V
Example 12: Describe: $g(x) = \frac{1}{2}|x - 4| + 3$
$a = \frac{1}{2}$: Vertical compression (wider V)
$h = 4$: Shift RIGHT 4 units
$k = 3$: Shift UP 3 units
Vertex: $(4, 3)$
Opens: Upward
Shape: Wide V
$a = \frac{1}{2}$: Vertical compression (wider V)
$h = 4$: Shift RIGHT 4 units
$k = 3$: Shift UP 3 units
Vertex: $(4, 3)$
Opens: Upward
Shape: Wide V
Transformation Summary Table
| Transformation | Form | Effect | Example |
|---|---|---|---|
| Vertical Shift UP | $f(x) = |x| + k$ | Move up $k$ units | $|x| + 3$ → up 3 |
| Vertical Shift DOWN | $f(x) = |x| - k$ | Move down $k$ units | $|x| - 2$ → down 2 |
| Horizontal Shift RIGHT | $f(x) = |x - h|$ | Move right $h$ units | $|x - 5|$ → right 5 |
| Horizontal Shift LEFT | $f(x) = |x + h|$ | Move left $h$ units | $|x + 3|$ → left 3 |
| Reflection | $f(x) = -|x|$ | Flip over x-axis | $-|x|$ → opens down |
| Vertical Stretch | $f(x) = a|x|$, $|a| > 1$ | Narrower V | $3|x|$ → 3x steeper |
| Vertical Compression | $f(x) = a|x|$, $0 < |a| < 1$ | Wider V | $\frac{1}{2}|x|$ → wider |
Quick Reference: $f(x) = a|x - h| + k$
| Parameter | What It Does | How to Remember |
|---|---|---|
| $a$ |
• $|a| > 1$: stretch (narrow) • $0 < |a| < 1$: compress (wide) • $a < 0$: flip upside down | Affects steepness and direction |
| $h$ |
• Horizontal shift • Moves vertex left/right • OPPOSITE of sign! | $x - h$ = right, $x + h$ = left |
| $k$ |
• Vertical shift • Moves vertex up/down • Same as sign | $+ k$ = up, $- k$ = down |
| Vertex | $(h, k)$ | Point of the V |
Properties of Absolute Value Functions
| Property | Description |
|---|---|
| Shape | V-shape (or upside-down V) |
| Domain | Always all real numbers: $(-\infty, \infty)$ |
| Range |
• Opens up: $[k, \infty)$ • Opens down: $(-\infty, k]$ |
| Vertex | Minimum (if up) or maximum (if down) point |
| Symmetry | Symmetric about vertical line through vertex |
| Axis of Symmetry | $x = h$ (vertical line through vertex) |
Success Tips for Absolute Value Functions:
✓ Absolute value is always non-negative (≥ 0)
✓ Graph shape is always V (or upside-down V)
✓ Domain is ALWAYS all real numbers
✓ Range depends on vertex and direction
✓ Vertex form: $f(x) = a|x - h| + k$ with vertex at $(h, k)$
✓ Horizontal shift: OPPOSITE of sign inside bars
✓ Vertical shift: SAME as sign outside bars
✓ Negative $a$ flips the V upside down
✓ $|a| > 1$ makes it narrower; $|a| < 1$ makes it wider
✓ Always evaluate inside the bars first!
✓ Absolute value is always non-negative (≥ 0)
✓ Graph shape is always V (or upside-down V)
✓ Domain is ALWAYS all real numbers
✓ Range depends on vertex and direction
✓ Vertex form: $f(x) = a|x - h| + k$ with vertex at $(h, k)$
✓ Horizontal shift: OPPOSITE of sign inside bars
✓ Vertical shift: SAME as sign outside bars
✓ Negative $a$ flips the V upside down
✓ $|a| > 1$ makes it narrower; $|a| < 1$ makes it wider
✓ Always evaluate inside the bars first!