Curve Architect Slides Algebra II
CURVE
ARCHITECT
Designing Rational Functions through Systems of Equations
Today's Blueprint
Learning Targets
Identify how a, h, and k transform rational functions.
Build a system of equations using coordinate points.
Solve for missing parameters using the Elimination Method .
Essential Question
How can we "reverse engineer" a function when we only have a few data points?
WARM-UP: THE PARENT
Reviewing the reciprocal function: \( y = \frac{1}{x} \)
5 MINUTES
Quick Recall
Where is the Vertical Asymptote?
Where is the Horizontal Asymptote?
Which quadrants contain the graph?
What happens to \(y\) as \(x\) gets very large?
Sketch the graph on your worksheet now!
x y
Reference Grid
THE ANATOMY OF RATIONAL FUNCTIONS
y =
a
x - h
+ k
Parameter [a]
Controls vertical stretch , shrink, or reflection.
Parameter [h]
Defines the Vertical Asymptote . \( x = h \)
Parameter [k]
Defines the Horizontal Asymptote . \( y = k \)
PHASE 1: THE SETUP
Watching: 0:00 — 4:25
Guiding Question
How do coordinates help find parameters?
Embedded media
YOUR CHALLENGE
Find the function \( y = \frac{a}{x - 2} + k \) that passes through:
(0, 3)
(4, 7)
1
Substitute points to create two equations .
2
Clear the fractions like in the video.
PHASE 2: THE ELIMINATION
Watching: 4:25 — End
New Strategy
Eliminating "k" to solve for "a"
Embedded media
Architect's Debrief
Critical Thinking
Why were two points absolutely necessary to find the final equation? What would happen if we only had one point?
a = 4
Vertical Stretch
k = 5
Horizontal Asymptote
h = 2
Vertical Asymptote
Submit your worksheet before leaving!
Curve Architect Worksheet Curve Architect
Algebra II — Rational Functions & Systems
NAME:
DATE:
1
Warm-Up: The Parent Function
Sketch the parent function \( y = \frac{1}{x} \) on the grid. Label the Vertical Asymptote (VA) and Horizontal Asymptote (HA) .
VA:
HA:
2
Video Notes (0:00 — 4:25)
Guiding Question
How do the coordinates of the points help us find the equation?
\( h \)
Vertical Asymptote
\( k \)
Horizontal Asymptote
\( a \)
Stretch / Shrink
3
The Main Challenge
Find the function \( y = \frac{a}{x - 2} + k \) that passes through (0, 3) and (4, 7).
Step 1: Set up Equation A
Substitute point (0, 3)
Step 2: Set up Equation B
Substitute point (4, 7)
Step 3: Solve the System (Elimination)
Watch video 4:25—End for a hint!
FINAL VALUE OF [a]
a = _____
FINAL VALUE OF [k]
k = _____
4
Verification & Closure
Final Function Equation: y = ___________________________
Architect's Reflection:
Why were TWO points necessary? Could you have found both "a" and "k" with only one point? Explain.
Blueprint v1.0 — Property of Curve Architect Inc.
Curve Architect Tech Guide Tech Verification Guide
Verifying Rational Curves: TI-84 & Desmos
TI-84 Plus Series
Step 1: Input Equation
Press Y=
Use ALPHA + Y= to choose n/d (fraction template).
Step 2: Set Asymptotes
Note: The calculator doesn't draw asymptotes, but you can see them in the table.
Check 2nd + GRAPH (Table). Look for ERROR at your vertical asymptote.
Step 3: Verify Points
Press TRACE and type your X-value (e.g., 0 or 4 ). Verify the Y-value matches your coordinates.
Pro Tip: "Vertical Lines"
If your TI-84 shows a solid vertical line, it's trying to connect the dots across the asymptote. Go to MODE and change CONNECTED to DOT for a cleaner look.
Desmos Graphing
Step 1: Input Equation
Type y = a / (x - h) + k
Replace a, h, k with your calculated values.
Step 2: Plot Points
In a new line, type your coordinates in parentheses: (0, 3) and (4, 7).
Click Label to see if the curve passes exactly through the points.
Step 3: Graph Asymptotes
Add lines like x = 2 and y = 5. Click the gear icon to make them Dashed .
Pro Tip: Sliders
"Try typing the variables a, h, and k first. Desmos will offer to 'Add Sliders.' You can slide them to see exactly how the graph shifts and stretches in real-time!"
Visual Verification Blueprint
Sketch View
Calc Interface
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Curve Architect Teacher Guide Teacher Facilitation Guide
Lesson: Curve Architect (Rational Functions)
45-50 MIN
HS ALGEBRA II
Pacing & Flow
00-05m
Warm-Up: The Parent Function
Students sketch \( y = 1/x \). Ensure they identify VA at \( x=0 \) and HA at \( y=0 \).
05-15m
Video Viewing: The Setup
Watch 0:00 — 4:25. Pause at 2:21 to discuss the strategy: "How many points do we need if we have two unknowns (a and k)?"
15-35m
Main Activity: The Challenge
Students work in pairs to set up Equation A and B. They should clear fractions before watching the rest of the video.
35-45m
Video Phase 2 & Verification
Watch 4:25 — End. Students apply the elimination method to their problem and use the Tech Guide to verify.
45-50m
Closure: Architect's Debrief
Review the reflection question. Emphasize that \( n \) unknowns require \( n \) independent equations.
Worksheet Answer Key
Warm-Up
VA: \( x = 0 \)
HA: \( y = 0 \)
Graph should show hyperbola in Q1 and Q3.
Guiding Question Answer
Coordinates provide specific \( (x, y) \) values that turn the function into a linear equation where \( a \) and \( k \) are the only unknowns. Using two points creates a system of two equations.
Main Challenge Solution
Equation A (Point 0,3):
3 = a / (0 - 2) + k
3 = -a/2 + k
-6 = a - 2k (Multiplied by -2)
Equation B (Point 4,7):
7 = a / (4 - 2) + k
7 = a/2 + k
14 = a + 2k (Multiplied by 2)
Elimination:
(-6) + 14 = (a - 2k) + (a + 2k)
8 = 2a \(\rightarrow\) a = 4
Solve for k:
14 = 4 + 2k \(\rightarrow\) 10 = 2k \(\rightarrow\) k = 5
Final Answer:
y = 4 / (x - 2) + 5
Common Architect Errors
Forgetting the 'k': Students often forget to multiply the 'k' term when clearing the fraction. (e.g., multiplying by 2 and getting \( 14 = a + k \) instead of \( 14 = a + 2k \)).Asymptote Signs: Remind students that the formula is \( x - h \). If the VA is at \( x = 2 \), the denominator is \( x - 2 \).Elimination Signs: Watch for students who subtract the equations incorrectly. Adding is usually safer once terms are balanced.
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