Lesson 2.17
Turning Points
Polynomial graphs have hills and valleys — called local maximums and local minimums. A degree- polynomial has at most turning points.
Introduction
A turning point is where the graph changes direction — from going up to going down (a local max) or from going down to going up (a local min). Knowing the maximum possible number of turning points helps you predict the overall shape of any polynomial.
Past Knowledge
End behavior (Lessons 2.2-2.4) and zeros/multiplicity (Lessons 2.15-2.16).
Today's Goal
Determine the maximum number of turning points, identify local extrema, and distinguish them from the absolute max/min.
Future Success
In calculus, you'll find exact turning points using derivatives. For now, we estimate using the degree and graph behavior.
Key Concepts
The Turning Point Rule
A degree- polynomial has at most turning points
The actual number of turning points can be fewer than , but never more. The difference is always even.
Quick Reference
| Degree | Max Turning Pts | Example |
|---|---|---|
| 2 | 1 | Parabola (1 vertex) |
| 3 | 2 | S-curve (1 max + 1 min) |
| 4 | 3 | W-shape possible |
| 5 | 4 | Two hills + two valleys |
Local vs. Absolute
Local Maximum
Higher than nearby points — a "hilltop" in the graph. There may be higher points elsewhere.
Local Minimum
Lower than nearby points — a "valley" in the graph. There may be lower points elsewhere.
Absolute Max/Min
The highest or lowest point overall. Odd-degree polynomials have no absolute max or min (they go to ).
Worked Examples
Example 1: Max Turning Points from Degree
BasicFor each polynomial, state the maximum number of turning points:
Degree: 5
At most 4 turning points
Degree: 4
At most 3 turning points
Degree: 7
At most 6 (but actually has 0)
💡 shows that the actual number can be much less than the maximum. It has no turning points at all — it's always increasing!
Example 2: Identifying Extrema from a Table
IntermediateThe table shows selected values of a degree-4 polynomial. Identify approximate turning points.
| −3 | −2 | −1 | 0 | 1 | 2 | 3 | |
| 10 | −2 | 3 | 8 | 5 | −1 | 12 |
Near : goes 10 → −2 → 3 (decreases then increases) → local minimum near
Near : goes 3 → 8 → 5 (increases then decreases) → local maximum near
Near : goes 5 → −1 → 12 (decreases then increases) → local minimum near
3 turning points (which is exactly ✓)
Example 3: Minimum Degree from Turning Points
AdvancedA polynomial graph has 5 turning points. What is the minimum possible degree?
Apply the Rule in Reverse
If turning points = at most, then .
But Parity Matters!
The difference between degree and actual turning points is always even. So a degree-6 polynomial could have 5, 3, or 1 turning points — 5 is allowed.
Minimum degree: 6
Common Pitfalls
Saying "Exactly Turning Points"
The rule gives a maximum, not a guarantee. is degree 5 but has zero turning points, not 4.
Confusing Local and Absolute
A local max is the highest nearby — not necessarily the highest overall. Odd-degree polynomials go to , so they have no absolute max or min.
Real-Life Applications
In economics, revenue and cost curves are often polynomial. Turning points represent break-even points and profit peaks. Knowing how many turning points a model can have tells analysts how many times the market can shift direction within the modeled period.
Practice Quiz
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