AP Precalculus Section 1.6 Example: End Behavior in Limit Notation of a Polynomial (Example 2)

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Describing the end behavior of a polynomial function using limit notation involves understanding how the function behaves as \(x\) approaches positive or negative infinity. For a polynomial function of the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where \(n\) is the degree of the polynomial and \(a_n\) is the leading coefficient:

### Steps to Write End Behavior in Limit Notation for a Polynomial:

1. **Determine the Leading Term**:
- Identify the term in the polynomial with the highest power of \(x\). This term dominates the behavior of the polynomial as \(x\) approaches infinity.

2. **Analyze the Leading Term**:
- Determine the behavior based on the leading term:
- If the leading term has an even power (\(n\) is even):
- For \(a_n greater than 0\), the end behavior as \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\) is:
\(\lim_{{x \to +\infty}} f(x) = \lim_{{x \to -\infty}} f(x) = +\infty\)
- For \(a_n less than 0\), the end behavior as \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\) is:
\(\lim_{{x \to +\infty}} f(x) = \lim_{{x \to -\infty}} f(x) = -\infty\)
- If the leading term has an odd power (\(n\) is odd):
- For both \(a_n greater than 0\) and \(a_n less than 0\), the end behavior as \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\) is:
\(\lim_{{x \to +\infty}} f(x) = \lim_{{x \to -\infty}} f(x) = \pm \infty\)

### Example:

Consider the polynomial function \(f(x) = 3x^4 - 2x^3 + 5x - 1\).

1. **Determine the Leading Term**:
- The leading term is \(3x^4\) since it has the highest power of \(x\).

2. **Analyze the Leading Term**:
- The degree (\(n\)) is even (\(4\)).
- The leading coefficient (\(a_n\)) is positive (\(3\)).
- End behavior as \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\) is:
\(\lim_{{x \to +\infty}} f(x) = \lim_{{x \to -\infty}} f(x) = +\infty\)

### Conclusion:

By focusing on the leading term and considering its power and coefficient, you can determine the end behavior of a polynomial function using limit notation as \(x\) approaches positive or negative infinity.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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