AP Precalculus Section 1.6 Example: Write a Polynomial from a Graph

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

To identify a polynomial on a multiple-choice exam using end behavior and the number of local extrema, follow these steps:

1. Determine the degree of the polynomial: Look at the highest power of the variable in the expression. This tells you the degree of the polynomial.

2. Analyze the end behavior:
- If the degree is even and the leading coefficient is positive, both ends of the graph will rise or fall in the same direction.
- If the degree is even and the leading coefficient is negative, one end rises while the other falls.
- If the degree is odd and the leading coefficient is positive, one end rises while the other falls.
- If the degree is odd and the leading coefficient is negative, both ends rise in opposite directions.

3. Count the number of local extrema: Local extrema are points where the graph changes from increasing to decreasing or vice versa. Count the number of peaks (local maxima) and valleys (local minima) on the graph.

4. Use this information to eliminate answer choices:
- If the end behavior and the number of local extrema match the given options, you've likely identified the correct polynomial.
- Eliminate answer choices that don't match the observed behavior.

For example, if you're given a polynomial graph that rises on both ends and has two local maxima and one local minimum, you can deduce that it's likely a cubic polynomial with a positive leading coefficient. Use this information to identify the correct answer choice on the exam.

Remember that while this approach can help narrow down your choices, it's essential to consider other characteristics of the graph, such as the behavior near the x-intercepts (roots), to confirm your answer.

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
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