Function Blueprints Slides Function Blueprints
Reverse Engineering Rational Graphs
Pre-Calculus
Unit: Rational Functions
Mission
"By the end of this session, you will construct equations for rational functions given their visual blueprints (graphs)."
Project Timeline
01
Warm-up: Quick-Fire Shifts
5 minutes on whiteboards
02
Briefing: Shifting Rational Graphs
10 minutes video analysis
03
Main Activity: Graph Detective
25 minutes pair investigation
04
Extension: Reflection Challenge
5 minutes wrap-up
Warm-up: Quick-Fire Shifts
Get those whiteboards ready!
Rules of the Game:
The teacher will call out a translation (e.g., "Left 4").
You write the change to the equation on your board (e.g., \(x + 4\)).
Hold up your board when you hear "Flip!"
Wait for the Teacher's Command
Remember: Horizontal shifts are counter-intuitive !
\(f(x-h) + k\)
Case Study Analysis
Watch: Working Backwards (10:33 - 12:19)
Embedded media
LOOK FOR
How the vertical and horizontal asymptotes map to the equation constants \(a\) and \(b\).
NOTE
How a specific point is used to verify the final equation blueprint.
Graph Detective
A
The Architect
Draw a shifted reciprocal graph with integer asymptotes and one clearly marked integer point .
B
The Detective
Determine the horizontal & vertical asymptotes, write the equation, and prove it works using the point.
Evidence Requirements
Must include horizontal asymptote \(y = k\)
Must include vertical asymptote \(x = h\)
Must label at least one integer point \((x, y)\)
Switch roles after 10 minutes!
Extension Challenge
What if we reflect the graph across the x-axis?
Visual Change
What would happen to the curve's direction? Which quadrants (relative to the asymptotes) would it occupy?
Algebraic Change
How would you modify the equation \(f(x) = \frac{1}{x-h} + k\) to reflect this change?
Be prepared to share your hypothesis with the class.
Graph Detective Worksheet Graph Detective
Mission: Reverse-Engineer the Equation
Agent Name:
Date:
Partner:
01
Case #1: You are the Architect
Instructions: On the grid provided, draw a shifted reciprocal function graph.integer values .one specific integer point $(x, y)$ that lies on the curve.
Secret Blueprint (Don't let partner see!)
H.A. Equation:
V.A. Equation:
x
y
02
Case #2: You are the Detective
Exchange papers with your partner. Analyze their graph to find the hidden equation.
Evidence: Visual Analysis
Vertical Asymptote:
x =
Horizontal Asymptote:
y =
Given Point:
( , )
The Final Equation Blueprint
f(x) =
Verification (Show your algebra)
Plug the given point $(x, y)$ into your equation to prove it's correct.
Reflecting on the Case
Why is it critical to have a specific point on the curve even after you find the asymptotes? What part of the equation might change if the graph was "stretched" vertically (even if the asymptotes stayed the same)?
Quick Draw Coordinate Planes Quick-Draw Coordinate Plane
Personal Whiteboard Template
x
y
f(x) =
Quick-Draw Coordinate Plane
Personal Whiteboard Template
x
y
f(x) =
Case File Teacher Guide Case File: Function Blueprints
Teacher Facilitation Guide
Subject: Pre-Calculus
Duration: 50 Minutes
Lesson Objective
Students will "reverse engineer" rational functions of the form $f(x) = \frac{1}{x-h} + k$ by identifying vertical and horizontal asymptotes from a graph and verifying their algebraic model with a specific point.
Key Materials
• Blueprint Slides
• Graph Detective Worksheets
• Quick-Draw Planes (in dry-erase sleeves)
• Dry-erase markers
01: Warm-up: Quick-Fire Shifts (5 min)
Use the Quick-Draw Planes . Say the prompt, give 10 seconds, then call "FLIP!"
Teacher Says:
"Horizontal Shift: Right 5"
"Vertical Shift: Down 2"
"Shift: Up 10, Left 3"
"The Vertical Asymptote is x = -4"
Students Write:
(x - 5) in denominator
-2 at end of function
1/(x+3) + 10
(x + 4) in denominator
02: Video Analysis (10 min)
Focus: Example 3 (10:33 - 12:19)
!
Pause at 10:33: Before Justin starts, ask: "Just by looking, what are the two most obvious 'landmarks' on this graph?" (Ans: The red dashed lines/asymptotes).
!
Key Point: Emphasize that $b$ (the vertical shift) is the Horizontal Asymptote, and $a$ (the horizontal shift) is the Vertical Asymptote. Note the sign change for $a$.
03: Graph Detective (25 min)
Phase 1: The Architect (7 min)
Students draw their graphs. Circulate and ensure they pick integer points . A common error is drawing a curve that doesn't actually hit an integer coordinate. Suggest they start with the point (e.g., 3, 5) then place asymptotes relative to it (e.g., x=2, y=4).
Phase 2: The Detective (10 min)
Students swap. They must write the equation. If they get stuck, ask: "Which line tells you what value to add at the end? Which line tells you what to put next to x?"
04: Extension Solution & Closing
Challenge: Reflection across the x-axis.
The Change:
Multiplying the entire function by -1.
Visual Result:
The branches flip. Instead of being in the Top-Right/Bottom-Left of the asymptotes, they move to Top-Left/Bottom-Right.
Troubleshooting the Mission
