Morphing Models Teacher Guide Morphing Models
Function Transformations • 11th-12th Grade Pre-Calculus
Lesson Guide
Duration: 45-50 Minutes
Learning Objective
Students will predict how algebraic changes to a function equation (shifts, reflections, stretches, and shrinks) affect its graph and verify their predictions using graphing technology.
Materials Needed
Shift Logic Slide Deck
Transformation Detectives Worksheet
Graphing Calculators (TI-84 or Desmos)
Function Pilot Guide (Reference Sheet)
Instructional Sequence
5 MIN
Warm-Up: The Baseline
Students graph the parent function \(y = x^2\) on their worksheet. They then predict the appearance of \(y = x^2 + 2\). Ask: "Does the '+2' happen before or after the squaring?"
15 MIN
Video Investigation
Watch the "Transformations" segment (18:24 - 29:17). Use the pauses indicated in the slide deck to let students predict transformations for:
\(y = -x^2 + 2\)
\(y = -|x - 2| + 2\)
20 MIN
Transformation Detectives
Students work individually or in pairs to "crack" complex equations. They must list the transformations in the correct order (Horizontal Shift → Stretch/Shrink/Reflect → Vertical Shift) before graphing to verify.
5 MIN
The "Hardest to Spot" Reflection
Class discussion: "Which transformation was hardest to identify?" Focus on horizontal shifts with internal negatives, like \(f(3-x)\).
Case File: Answer Key
y = -1/(x + 3)^2 - 2
Order: Left 3 units, Reflect over x-axis, Down 2 units.
y = 3|x - 5| + 1
Order: Right 5 units, Vertical Stretch by 3, Up 1 unit.
y = \sqrt{-(x + 1)} + 4
Order: Left 1 unit, Reflect over y-axis, Up 4 units.
y = 3 - \sqrt{5 - x}
CRITICAL CASE: Factor first → \(y = -\sqrt{-(x - 5)} + 3\)
Order: Right 5 units, Reflect over x-axis, Reflect over y-axis, Up 3 units.
Morphing Models Slides Morphing Models
The Logic of Function Transformations
Warm-Up: Predictive Plotting
5 Minutes
On your worksheet:
Sketch the parent: y = x²
Predict the transformation: y = x² + 2
Prediction Logic
Where does the +2 move the graph? Why?
Video Investigation
Embedded media
Watch for the logic behind shifts and reflections.
18:24 - 29:17
Pause & Predict
Target 1
y = -x² + 2
Does the reflection or the shift happen first? Sketch your guess.
Target 2
y = -|x - 2| + 2
Three transformations here. What's the order of operations?
Mission: Transformation Detectives
01
ANALYZE the equation without looking at a graph.
02
LIST every transformation in order (Horizontal → Stretch → Vertical).
03
VERIFY your "case file" using your graphing calculator.
The Final Verdict
"Which transformation is the hardest to spot in an equation? What makes it so elusive?"
Pro Tip: Beware of reflections inside the function!
Transformation Detectives Worksheet Transformation Detectives
Field Notes: Algebraic Function Investigation
NAME:
DATE:
Warm-Up
1. Sketch \(y = x^2\) (the parent).
2. Predict & Sketch \(y = x^2 + 2\).
Case Evidence: Video Log
Trial A: \(y = -x^2 + 2\)
Describe the movement/flip:
Trial B: \(y = -|x - 2| + 2\)
Describe the movement/flip:
The Investigation: Advanced Transformations
Analyze each equation. List transformations in the order they occur. Then, graph on your calculator to verify.
y = \frac{-1}{(x + 3)^2} - 2
y = 3|x - 5| + 1
y = \sqrt{-(x + 1)} + 4
The Ultimate Case
y = 3 - \sqrt{5 - x}
Show Work (Factor first!):
Final Verdict (Transformations):
Reflection: The Elusive Shift
Which of these was hardest to spot correctly? Why do horizontal shifts with negative coefficients (like Case 4) cause the most confusion?
Function Pilot Guide Reference Sheet Function Pilot Guide
Navigational Reference for Function Transformations
The Transformation Blueprint
y = a f(b(x - h)) + k
a
Vertical Adjustments (Outside)
Happens AFTER the function operation.
|a| > 1: Vertical Stretch
0 < |a| < 1: Vertical Shrink
a is Negative: Reflect over x-axis
b
Horizontal Adjustments (Inside)
Happens BEFORE the function operation. (Opposite!)
|b| > 1: Horizontal Shrink (by factor 1/b)
0 < |b| < 1: Horizontal Stretch (by factor 1/b)
b is Negative: Reflect over y-axis
h
Horizontal Shift (Inside)
(x - h): Shift RIGHT h units
(x + h): Shift LEFT h units
Tip: Watch for (b(x-h)) factoring!
k
Vertical Shift (Outside)
+ k: Shift UP k units
- k: Shift DOWN k units
Calculator Verification
Step 1
Enter the Parent Function into Y1 (e.g., \(Y1 = x^2\)).
Step 2
Enter the Transformed Function into Y2.
Step 3
Change the line style for Y2 to Bold or a different color to distinguish it.
Step 4
Press GRAPH and use TRACE to compare points.
Order of Operations
To graph correctly by hand, apply transformations in this sequence:
01 Horizontal Shifts (h)
02 Horizontal Shrink/Stretch (b)
03 Reflections (a, b)
04 Vertical Stretch/Shrink (a)
05 Vertical Shifts (k)
CRITICAL WARNING: When there is a value for b (inside the function), you MUST factor it out to find the true horizontal shift.Right 3 , even though it looks like \( \sqrt{-x+3} \).
