Vertex Voyage Anchor Chart Vertex Voyage Anchor Chart
Navigating Absolute Value Transformations: \(y = a|x - h| + k\)
The Base Camp
The Parent Function: \(f(x) = |x|\)
Shape: Symmetric "V"Vertex: \((0, 0)\)Domain: \((-\infty, \infty)\)Range: \([0, \infty)\)
The Navigation Tool
The Absolute Equation
y = a|x - h| + k
a
Stretch, Compress, Flip
\(|a| > 1\): Stretch. \(|a| < 1\): Compression. Negative (\(a < 0\)): Opens down.
h
Horizontal Shift (Vertex X)
Moves left/right. Inverted! \(|x - 3|\) moves RIGHT 3. \(|x + 5|\) moves LEFT 5.
k
Vertical Shift (Vertex Y)
Moves up/down with the sign. \(+2\) moves UP 2; \(-4\) moves DOWN 4.
Transformation Log
Translation & Scaling Guide
Shift Equation Visual Effect Right y = x - h Left y = x + h Up y = x Down y = x Stretch y = a x Shrink y = a x Flip y = - x
Step-By-Step Voyage
Coordinate Map & Voyage Steps
Parent Point
\((x, y)\)
Transformed Point
\((x + h, ay + k)\)
1
Find the Vertex: Plot the point \((h, k)\). Remember to invert the sign of \(h\) from inside the bars!
2
Find Direction: If \(a < 0\), it opens down. If \(a > 0\), it opens up.
3
Plot Key Points: Use \(a\) as a "slope" to go right 1 unit and vertical by \(a\). Mirror on the left!
Example Expedition
Mapping: \(y = -2|x - 3| + 4\)
1. Find Vertex
Vertex \((h, k) = (3, 4)\).
Sign of h is inverted; k is kept.
2. Slope & Opening
\(a = -2\) (Opens Down)
Steepness multiplier of 2; flips over the x-axis.
3. Key Coordinates
\((0, 0) \rightarrow (3, 4)\)
\((1, 1) \rightarrow (4, 2)\)
\((-1, 1) \rightarrow (2, 2)\)
Algebra 1 • Vertex Voyage: Absolute Value Transformations Name: _______________________________ Date: _______________
Vertex Voyage Presentation Slides Vertex Voyage
Algebra Expedition
Navigating Absolute Value Transformations
Discover the mathematical compass to shift, stretch, and reflect the absolute value function. Master vertex form and chart your coordinates!
Welcome Aboard
Press Next to Set Sail
The Base Camp
Slide 2 of 6
The Parent Function: \(f(x) = |x|\)
Every voyage begins at home. The absolute value function creates a symmetrical V-shape centered at the origin.
Key Properties
• Vertex Location: (0, 0)
• Domain: \((-\infty, \infty)\)
• Range: \([0, \infty)\)
Key Points
• \((-1, 1)\)
• \((0, 0)\)
• \((1, 1)\)
The Parent Function Vertex Voyage Expedition
Translations
Slide 3 of 6
Inside the bars
Horizontal Shift: \(|x - h|\)
Shifts the graph Left or Right. This shift is the opposite of the sign!
\(y = |x - 4|\) shifts RIGHT 4 units
\(y = |x + 4|\) shifts LEFT 4 units
Outside the bars
Vertical Shift: \(|x| + k\)
Shifts the graph Up or Down. This shift follows the exact sign!
\(y = |x| + 3\) shifts UP 3 units
\(y = |x| - 3\) shifts DOWN 3 units
Vertex Location: \((h, k)\) Vertex Voyage Expedition
The Multiplier
Slide 4 of 6
\(|a| > 1\)
Vertical Stretch
Multiplies the y-coordinates, making the V-shape steeper and narrower.
Example: \(y = 3|x|\)
\(0 < |a| < 1\)
Vertical Compression
Multiplies the y-coordinates, making the V-shape wider and flatter.
Example: \(y = \frac{1}{2}|x|\)
\(a < 0\)
Vertical Reflection
Flips the V-shape over the x-axis, causing it to open downwards.
Example: \(y = -|x|\)
The Value of "a" Charts Slope & Direction Vertex Voyage Expedition
The Flight Plan
Slide 5 of 6
The Three-Step Voyage Checklist
Vertex Voyage Worksheet Vertex Voyage Handout
Absolute Value Transformations Practice
Name: ___________________________
Date: _________________ Period: _____
Your Objective
Chart your course through algebraic translations, reflections, and dilations. For each problem, analyze how the standard vertex formula \(y = a|x - h| + k\) shifts the parent function \(f(x) = |x|\).
Section A
Identify the Navigational Changes
List all individual transformations (shifts left/right/up/down, vertical stretch/compression, reflection) and identify the new vertex.
PROBLEM 1
y = |x - 4| + 7
Transformations:
Vertex: ___________________
PROBLEM 2
y = -3|x + 1| - 2
Transformations:
Vertex: ___________________
PROBLEM 3
y = \frac{1}{2}|x| - 5
Transformations:
Vertex: ___________________
PROBLEM 4
y = -0.75|x - 2| + 3
Transformations:
Vertex: ___________________
Section B
Write the Equation
Construct the transformed absolute value equation according to the visual coordinates provided.
5
Shifted 6 units right, 3 units down, and opens upwards with parent steepness.
y =
6
Reflected across the x-axis, shifted 2 units left, and 5 units up.
y =
7
Vertically compressed by a factor of 0.4, shifted 8 units right, and 1 unit up.
y =
8
Vertically stretched by a factor of 4, reflected across the x-axis, and shifted 9 units down.
y =
Vertex Voyage Workbook • Page 1 of 2 Algebra 1 • Unit 4
Vertex Voyage Handout
Absolute Value Transformations Practice
Name: ___________________________
Date: _________________ Period: _____
Section C
Chart & Graph Your Voyages
First, list the key values and characteristics. Then, carefully graph the transformed function on the coordinate grid provided.
EXPEDITION 9
y = 2|x - 3| - 4
Vertex (h, k)
Direction of V
Domain
Range
Tip: Plot the vertex first, then use \(a = 2\) as slope to navigate 1 unit right and 2 units up, mirroring on the left!
x y
EXPEDITION 10
Vertex Voyage Answer Key Teacher Answer Key
Vertex Voyage Key
Absolute Value Transformations Practice • Reference Guide
SOLUTIONS MANUAL
Standard Curriculum Pack
Teacher Instructions
Review student responses for coordinate signs. The horizontal translation value h is standardly written as x - h, which causes the direction of translation to invert. Check that students plotted vertices correctly on Page 2 before drafting the lines.
Section A
Identify the Navigational Changes - Solutions
PROBLEM 1
y = |x - 4| + 7
Transformations:
Shifted right 4 units, shifted up 7 units
Vertex: (4, 7)
PROBLEM 2
y = -3|x + 1| - 2
Transformations:
Shifted left 1, down 2; reflected (opens down); vertically stretched by 3
Vertex: (-1, -2)
PROBLEM 3
y = \frac{1}{2}|x| - 5
Transformations:
Shifted down 5 units; vertically compressed by \(\frac{1}{2}\)
Vertex: (0, -5)
PROBLEM 4
y = -0.75|x - 2| + 3
Transformations:
Shifted right 2, up 3; reflected (opens down); vertically compressed by 0.75
Vertex: (2, 3)
Section B
Write the Equation - Solutions
5
Shifted 6 units right, 3 units down, and opens upwards with parent steepness.
y =
6
Reflected across the x-axis, shifted 2 units left, and 5 units up.
y =
7
Vertically compressed by a factor of 0.4, shifted 8 units right, and 1 unit up.
y =
8
Vertically stretched by a factor of 4, reflected across the x-axis, and shifted 9 units down.
y =
Vertex Voyage Workbook Answer Key • Page 1 of 2 Algebra 1 • Solutions Guide
Teacher Answer Key
Vertex Voyage Key
Absolute Value Transformations Practice • Graphs Key
SOLUTIONS MANUAL
Standard Curriculum Pack
Section C
Chart & Graph Your Voyages - Graphed Solutions
EXPEDITION 9
y = 2|x - 3| - 4
Vertex (h, k)
(3, -4)
Direction of V
Opens Up
Domain
\((-\infty, \infty)\)
Range
\([-4, \infty)\)
