Frankenstein Functions Presentation Frankenstein Functions
Project: Piecewise Monster Construction
Pre-Calc Lab
Warm-up: The Spark
"The monster is alive, but its parts are... disconnected."
On your paper:
Sketch a graph with a Jump Discontinuity at x = 2.
Identify the left-hand and right-hand limits for your sketch.
5 Minutes Remaining
Anatomy Review
Embedded media
Holes
Limit Exists
Jumps
Limit DNE
Asymptotes
Limit → ∞
Your Monster Awaits
Lab Objectives:
Stitch together algebraic "body parts" to create a single Piecewise Monster that satisfies all DNA Requirements.
// DNA REQUIREMENTS:
Limit of 2 at x = 1
Jump Discontinuity at x = 3
Limit → ∞ at x = -2
Equipment:
Equation Cut-outs
Algebraic "Glue"
The Lab Rubric
The Gallery Walk
Leave your monster at your lab station. Walk the gallery and use your Safety Checklist to verify your peers' creations.
ANALYZING SPECIMEN CONTINUITY...
Frankenstein Functions Worksheet Frankenstein Functions
SUB-SECTION: PIECEWISE CONSTRUCTION LAB
Specimen:
Date:
PART I: THE SPARK (WARM-UP)
Sketch a graph with a Jump Discontinuity at \(x = 2\). Clearly label your left and right limits.
Left-Hand Limit:
Right-Hand Limit:
Does the overall limit exist?
PART II: THE MONSTER DNA (REQUIREMENTS)
Priority Alpha
// YOUR PIECEWISE FUNCTION MUST SATISFY THE FOLLOWING:
R1 The overall limit as \(x \to 1\) must exist and equal \(2\).
R2 There must be a Jump Discontinuity at \(x = 3\).
R3 The function must have an Infinite Discontinuity at \(x = -2\).
Part III: The Stitching Table
Choose pieces from the "Body Parts" sheet and determine their domains to satisfy the DNA above. Paste or write them below.
PART
ALGEBRAIC PIECE (EQUATION)
DOMAIN INTERVAL
Head
If
Torso
If
Limbs
If
The Specimen Portrait
GRID: 1 UNIT = 1 SQUARE
Final Piecewise Equation: \(f(x) =\)
Write your complete piecewise function here using curly bracket notation.
Gallery Walk Certification
TO BE COMPLETED BY AN INSPECTOR (CLASSMATE):
Does the limit at \(x=1\) approach 2 from BOTH sides?
Is there a visible "Jump" at \(x=3\)?
Does the function shoot to \(\pm\infty\) at \(x=-2\)?
Inspector Initials:
Lab Supply: Body Parts
Select the equations that will help you build your monster. You may use pieces multiple times or modify them by shifting.
\(y = x + 1\)
\(y = 2x - 1\)
\(y = 2\)
\(y = -3\)
\(y = x^2\)
\(y = -(x-1)^2 + 2\)
\(y = \frac{1}{x+2}\)
\(y = \frac{-1}{(x+2)^2}\)
\(y = |x - 1| + 2\)
\(y = |x| - 4\)
\(y = \sqrt{x+3}\)
\(y = \frac{x^2-1}{x-1}\)
Note: You are responsible for ensuring your choices don't create "accidental" discontinuities where you want the monster to be "connected"!
Frankenstein Functions Rubric Lab Evaluation Rubric
PROJECT: FRANKENSTEIN FUNCTIONS
Lead Scientist:
Lab Station:
Criteria Level 1: Error Level 2: Unstable Level 3: Viable Level 4: Masterpiece Limit at \(x=1\)
Requirements: \(\lim \to 2\)
| Limit does not exist or approaches wrong value. | Correct value from one side only; LHL \(\neq\) RHL. | Correct LHL and RHL; overall limit is 2. Minor notation error. | Perfect algebraic alignment and clear graphical evidence of limit 2. |
|
Jump at \(x=3\)
Requirements: LHL \(\neq\) RHL
| Function is continuous at \(x=3\) or has a hole. | Incorrectly placed or poorly defined algebraically. | Distinct jump visible on graph and defined in piecewise function. | Mathematically precise jump with correct use of open/closed circles on graph. |
|
Asymptote at \(x=-2\)
Requirements: Vertical Asymptote
| No infinite behavior present at \(x=-2\). | Rational piece used but asymptote is at the wrong x-value. | Vertical asymptote exists; graph clearly shows unbounded growth. | Strategic use of rational pieces to create clear asymptotic behavior. |
|
Graphing Accuracy
| Graph is messy or unrelated to equations. | Significant errors in translating equations to the coordinate plane. | Mostly accurate graph; pieces generally match their defined intervals. | Exemplary precision. All endpoints and curves are mathematically sound. |
|
Formal Notation
| Missing piecewise equation or curly bracket notation. | Equations or domain intervals contain multiple logic errors. | Complete piecewise notation with few errors in domain syntax. | Correct mathematical syntax for piecewise functions and intervals. |
Lab Supervisor Comments:
Final Lab Grade
____/20
"It's Alive!"
OFFICIAL LAB
Frankenstein Functions Teacher Guide Lab Supervisor Guide
Teacher Resource: Frankenstein Functions
Subject Context
This activity moves students from passive graph analysis to active construction. By "stitching" together function pieces, students must grapple with the algebraic requirements for continuity (meeting at a point) versus limits that do not exist (jumps/asymptotes).
Learning Objective
Students will construct a multi-part piecewise function that satisfies specific limit existance and failure criteria at defined x-values.
Prep Checklist
Print 1 Worksheet per student.
Check Scissors/Glue inventory.
Load Slide Presentation.
Review "Possible Solutions" (below).
Lab Timeline (50 Minutes)
00-05
The Spark (Warm-up)
Students sketch a jump discontinuity on Page 1. Circulate and check that they understand the concept of limits approaching different values.
05-10
Anatomy Review (Video)
Show the summary (13:10-end). Focus on the "Key Takeaways" slide to reinforce the vocabulary of discontinuities.
10-35
The Stitching (Activity)
Students cut and paste body parts. Encourage them to "test" their pieces on the coordinate plane before gluing. They must define the domain intervals to create the desired behavior.
35-45
Gallery Walk (Closure)
Students leave work on desks. Peer "Inspectors" use the checklist on Page 2 to certify the monsters. 3-4 rotations recommended.
45-50
Lab Debrief
Quickly discuss common pitfalls (e.g., creating a jump where a limit was supposed to exist).
Specimen Solution Example
While multiple monsters can be built, here is one "Viable Specimen":
// PIECE 1: ASYMPTOTE at x=-2
\(f(x) = \frac{1}{x+2}\) if \(x < -1\)
// PIECE 2: LIMIT 2 at x=1
\(f(x) = x + 1\) if \(-1 \le x < 3\)
Note: At x=1, f(1)=2. Limit from both sides matches.
// PIECE 3: JUMP at x=3
\(f(x) = -3\) if \(x \ge 3\)
Note: Approaches 4 from left, is -3 on right. JUMP!
Logic Breakdown
To satisfy Limit=2 at x=1 , the piece covering \(x=1\) must be continuous there or have a hole at that specific y-value.
To satisfy Jump at x=3 , the piece ending at 3 and the piece starting at 3 must approach different y-values.
To satisfy Asymptote at x=-2 , the domain of the rational piece must include -2.
