Reciprocal Shift Presentation Slides Algebra II: Unit 4
RECIPROCAL
SHIFT SHOWDOWN
Mastering vertical and horizontal translations of the parent function \(f(x) = \frac{1}{x}\).
Warm-up: Error Analysis
The Scenario
Student A is analyzing the function: \(y = \frac{1}{x+2}\) They claim the graph moves RIGHT 2 units.
Identify the Mistake
Why did Student A think it moved right? What is the actual movement of the graph?
Think & Share
How can we prove the direction of the shift by checking a single point?
Time: 5 Minutes
Instructional Insights
Viewing: 8:44 - 10:32
Embedded media
Look For:
Example 1: Selecting the correct left-shift equation.
Example 2: Extracting HA and VA from an equation.
Key Terminology
Reciprocal Asymptote Translation
THE TRANSFORMATION TOOLKIT
General form of a translated reciprocal function
\(f(x) = \frac{1}{x - h} + k\)
Vertical (\(k\))
Moves the graph UP (\(+k\)) or DOWN (\(-k\)).
Horizontal (\(h\))
Moves the graph LEFT (\(+h\) in denom) or RIGHT (\(-h\) in denom).
Pro-Tip: The sign in the denominator is ALWAYS opposite the direction of the shift!
RELAY RULES
Collaboration is the key to victory
01
Squad Up
Groups of 3. Grab your Challenge Card Set and Tracking Sheet.
02
Solve & Agree
Each card has 3 tasks. Every group member must agree on the answer before writing it down.
03
Check-In
When a card is complete, raise your hand for a quick teacher check before moving to the next round.
20 MINUTES
GROUPS OF 3
Debrief & Reflect
What was the hardest shift to identify?
Was it matching a graph, or extracting asymptotes from a messy equation?
Top 3 Most Common "Trap" Mistakes
Swapping the direction of horizontal shifts.
Mixing up the Vertical and Horizontal Asymptotes.
Forgetting that \(x\) only affects the Vertical Asymptote.
Relay Challenge Cards Shift Showdown: Relay Cards
Algebra II // Reciprocal Translations
Group Name: ______________
Date: ______________
Relay Instructions: Work with your group of 3. Solve all three tasks for each round. You must all reach a consensus on the answer before recording it on your Tracking Sheet. When a round is finished, call for a teacher check to "unlock" the next round.
Round 01
Task 1 Description to Equation
A reciprocal function is shifted 3 units right and 4 units up . Write its equation.
Task 2 Equation to Asymptotes
Identify the Vertical (VA) and Horizontal (HA) asymptotes for: \(f(x) = \frac{1}{x+5} - 2\)
Task 3 Graph Interpretation
A graph of \(y = \frac{1}{x}\) has been translated. It now has a vertical asymptote at \(x = 1\) and hasn't moved vertically. Write the new equation.
Round 02
Task 1 Description to Equation
Shift the parent function \(1/x\) 6 units left and 1 unit down . Write the equation.
Task 2 Equation to Asymptotes
Identify the VA and HA for the function: \(g(x) = \frac{1}{x} + 8\)
Task 3 Graph Interpretation
A graph shows asymptotes at \(x = -2\) and \(y = 3\). If this is a translation of \(1/x\), what is the equation?
Round 03
Task 1 Description to Equation
Shift the parent function \(1/x\) 10 units to the right . Write the equation.
Task 2 Equation to Asymptotes
Identify the VA and HA for the function: \(h(x) = \frac{1}{x-7} - 5\)
Task 3 Graph Interpretation
A graph shows a vertical asymptote crossing the x-axis at \(4\) and a horizontal asymptote crossing the y-axis at \(-6\). Write the equation.
Round 04
Task 1 Description to Equation
A transformation includes a vertical shift down 9 units and a horizontal shift left 2 units . Write the equation.
Task 2 Equation to Asymptotes
Identify the VA and HA for the function: \(f(x) = \frac{1}{x+12} + 0.5\)
Task 3 Graph Interpretation
A graph's asymptotes intersect at the point \((-5, -5)\). Assuming a parent function of \(1/x\), write the equation.
Relay Tracking Sheet
Official Group Record
01
Task 1: Equation
Task 2: Asymptotes
Task 3: Match Graph
02
Task 1: Equation
Task 2: Asymptotes
Task 3: Match Graph
03
Task 1: Equation
Task 2: Asymptotes
Task 3: Match Graph
04
Task 1: Equation
Task 2: Asymptotes
Task 3: Match Graph
Relay Challenge Answer Key Shift Showdown: Answer Key
Teacher Resource // Relay Activity
Confidential
Grading Note: For Task 2, students should provide equations for the asymptotes (e.g., \(x = 5\) and \(y = 2\)). If they only provide the numbers, prompt them to use proper line notation. For Task 1 and 3, any equivalent mathematical notation is acceptable (e.g., \(f(x)\) vs \(y\)).
Round 01 Solutions
RECIPROCAL-KEY-R1
Task 1: Equation
\(y = \frac{1}{x-3} + 4\)
Task 2: Asymptotes
VA: \(x = -5\)
Task 3: Graph
\(y = \frac{1}{x-1}\)
Round 02 Solutions
RECIPROCAL-KEY-R2
Task 1: Equation
\(y = \frac{1}{x+6} - 1\)
Task 2: Asymptotes
VA: \(x = 0\)
Task 3: Graph
\(y = \frac{1}{x+2} + 3\)
Round 03 Solutions
RECIPROCAL-KEY-R3
Task 1: Equation
\(y = \frac{1}{x-10}\)
Task 2: Asymptotes
VA: \(x = 7\)
Task 3: Graph
\(y = \frac{1}{x-4} - 6\)
Round 04 Solutions
RECIPROCAL-KEY-R4
Task 1: Equation
\(y = \frac{1}{x+2} - 9\)
Task 2: Asymptotes
VA: \(x = -12\)
Task 3: Graph
\(y = \frac{1}{x+5} - 5\)
Common Pitfalls to Monitor
Sign Errors
Students often forget the \(h\) value is subtracted in the general form. Watch for \(y = \frac{1}{x+3}\) being labeled as a right shift instead of a left shift.
Asymptote Confusion
Ensure students associate the vertical shift (\(k\)) with the Horizontal Asymptote (\(y = k\)) and the horizontal shift (\(h\)) with the Vertical Asymptote (\(x = h\)).
