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Polynomial Function Graphs & End Behavior
AP Precalculus • Unit 1
Polynomial Function Graphs & End Behavior
Learn how the degree and leading coefficient of a polynomial determine its overall graph shape, end behavior, and number of turning points.
Lesson Objectives
- Identify end behavior of polynomial functions
- Understand how degree affects graph shape
- Relate leading coefficient to direction of the graph
- Estimate turning points of polynomials
1. Understanding End Behavior
The end behavior of a polynomial function describes what happens to the graph as
x → +∞ and x → −∞.
End behavior depends only on the leading term of the polynomial.
Key Idea:
As x becomes very large (positive or negative), the highest power of x dominates all other terms.
As x becomes very large (positive or negative), the highest power of x dominates all other terms.
📌 Diagram placeholder: dominance of leading term (OpenStax Fig. 3.29 style)
2. End Behavior Based on Degree
The degree of a polynomial determines whether both ends go in the same direction or in opposite directions.
Even Degree
Both ends move in the same direction.
Both ends move in the same direction.
- x², x⁴, x⁶
Odd Degree
Ends move in opposite directions.
Ends move in opposite directions.
- x³, x⁵, x⁷
📊 Diagram placeholder: even vs odd degree end behavior (OpenStax Fig. 3.30 style)
3. Effect of the Leading Coefficient
The sign of the leading coefficient determines whether the graph opens upward or downward.
Example (from the book):
f(x) = 2x³ → right end goes up, left end goes down
g(x) = −2x³ → right end goes down, left end goes up
📌 Diagram placeholder: positive vs negative leading coefficient
4. Turning Points of Polynomial Graphs
A turning point occurs where a graph changes direction from increasing to decreasing or vice versa.
Important Rule:
A polynomial of degree n can have at most n − 1 turning points.
A polynomial of degree n can have at most n − 1 turning points.
Example (OpenStax style):
A degree 4 polynomial can have at most 3 turning points.
A degree 4 polynomial can have at most 3 turning points.
📈 Diagram placeholder: polynomial with multiple turning points
5. Sketching Polynomial Graphs (Conceptual Steps)
- Identify the degree and leading coefficient
- Determine end behavior
- Find possible turning points
- Refine sketch using intercepts (next lesson)
Practice Questions
- Describe the end behavior of f(x) = −3x⁴
- How many turning points can a degree 5 polynomial have?
- Which way does the graph of f(x) = 7x³ open?
✅ Show Answer Key
- Both ends go down
- At most 4 turning points
- Right end up, left end down
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