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Transformations of Polynomial Functions
AP Precalculus • Unit 1
Zeros, Multiplicity & Graphical Behavior
Understand how zeros and their multiplicities determine whether a polynomial graph crosses, touches, or flattens at the x-axis.
Lesson Objectives
- Define zeros of polynomial functions
- Understand multiplicity of zeros
- Predict graph behavior at x-intercepts
- Link algebraic factors to graphical features
1. What Are Zeros of a Polynomial?
A zero of a polynomial function is a value of x for which:
f(x) = 0
Zeros correspond to the x-intercepts of the graph.
Example (OpenStax style):
If f(x) = (x − 2)(x + 1), then the zeros are:
If f(x) = (x − 2)(x + 1), then the zeros are:
- x = 2
- x = −1
📌 Diagram placeholder: polynomial graph crossing x-axis at zeros
2. Multiplicity of a Zero
The multiplicity of a zero tells us how many times a factor appears.
Definition:
If (x − a)k is a factor of the polynomial, then x = a is a zero of multiplicity k.
If (x − a)k is a factor of the polynomial, then x = a is a zero of multiplicity k.
Example (from the book):
f(x) = (x − 1)²(x + 3)
f(x) = (x − 1)²(x + 3)
- x = 1 has multiplicity 2
- x = −3 has multiplicity 1
📊 Diagram placeholder: repeated factor vs single factor
3. Graphical Behavior at Zeros
The behavior of the graph at a zero depends on whether the multiplicity is odd or even.
Odd Multiplicity
- Graph crosses the x-axis
- Examples: multiplicity 1, 3
Even Multiplicity
- Graph touches and turns
- Examples: multiplicity 2, 4
📈 Diagram placeholder: crossing vs touching behavior
4. Flattening at Higher Multiplicity
When the multiplicity is greater than 1, the graph becomes flatter near the zero.
Example (OpenStax style):
f(x) = (x − 2)³
f(x) = (x − 2)³
- Zero at x = 2
- Multiplicity = 3
- Graph crosses but flattens
📌 Diagram placeholder: flattening at zero
5. Summary Table: Multiplicity & Behavior
| Multiplicity | Behavior at Zero |
|---|---|
| Odd | Crosses x-axis |
| Even | Touches and turns |
| Greater than 1 | Flattens near zero |
Practice Questions
- Find the zeros and multiplicities of f(x) = (x − 3)²(x + 2)
- Will the graph cross or touch the x-axis at x = 3?
- Describe the behavior at x = −2
✅ Show Answer Key
- x = 3 (multiplicity 2), x = −2 (multiplicity 1)
- Touches and turns
- Crosses the x-axis
© Aviate Learning – AP Precalculus