Turning Point Slides
Case File #11-2026
Turning Point Case
The Mystery of the Excluded Vertex
11th Grade Pre-Calculus
Unit 2: Function Analysis
Warm-up: Vertex Analysis
Consider the function \( f(x) = -x^2 + 4 \).
What is the slope exactly at the peak (vertex) of this parabola?
Think about it:
- Is it going up?
- Is it going down?
- Is it doing something else entirely?
Exhibit A: Parabolic Vertex
Key Evidence
Focus: 4:02 - 4:30
Embedded media
The Definition
"At this one point... the slope is neither increasing nor decreasing. It's completely horizontal."
The Neutral Zone
"That point doesn't fit... so we exclude it from both intervals by using parentheses."
The Open vs. Closed Debate
Strictly Increasing
For \( a < b \), then \( f(a) < f(b) \).
Requires Parentheses \( ( ) \)
This is our standard for Calculus preparation.
Monotonically Increasing
For \( a < b \), then \( f(a) \leq f(b) \).
Allows Brackets \( [ ] \)
Discussion Anchors:
If the slope at the vertex is zero, can we truly say the function is "going up" at that exact spot?
Why might some textbooks disagree? (Hint: Think about the "or equal to" definition).
Function Forensics
Smooth Turns vs. Sharp Turns
Does the "Zero Slope" rule always apply?
Smooth Turn
f(x) = x²
Slope = 0
Sharp Turn (Cusp)
f(x) = |x|
Slope = Undefined
The Verdict:
Whether slope is zero OR undefined at a turning point, we still exclude it from strictly increasing/decreasing intervals.
Closing Statement
Intervals describe a span of inputs, not a single snapshot.
At a turning point, the function is in neutral.
In Pre-Calculus, we use parentheses ( ) to exclude these horizontal pauses.
Vertex Verdict Exit Ticket
The Vertex Verdict
Case File Summary: #11-2026
Lead Investigator (Name)
Date
Exhibit A: The Turning Point
At the exact point highlighted above, the function transitions from increasing to decreasing. The instantaneous rate of change is zero.
The Verdict: Final Justification
Explain in one clear sentence why we use parentheses ( ) instead of brackets [ ] when writing an interval that ends or begins at the vertex of a parabola.
Case Verification Checklist
Mentioned slope or derivative?
Addressed "zero" slope?
Reference to "neutral" zone?
Clear and concise?
Verified Pre-Calc Investigation
