AP Precalculus Section 1.4 Example: Absolute Extrema and Relative Extrema 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!

When analyzing a graph to label absolute (global) and relative (local) extrema, you're identifying the highest (maximum) or lowest (minimum) points of a function within different scopes: across the entire domain for absolute extrema and within specific intervals for relative extrema.

### Absolute Extrema:
- **Definition**: Absolute extrema are the highest or lowest points of the entire function over its entire domain.

#### Steps to Identify Absolute Extrema from a Graph:
1. **Scan the Entire Graph**:
- Observe the highest points (peaks) and lowest points (valleys) across the entire graph.

2. **Check End Behavior**:
- Note the behavior of the graph at the ends of its domain (left and right sides).
- Absolute extrema might occur at the endpoints if the function is bounded.

3. **Identify Absolute Extrema**:
- Mark the highest and lowest points on the graph. These represent the absolute maximum and minimum values of the function over its domain.

### Relative Extrema:
- **Definition**: Relative extrema are the highest (local maximum) or lowest (local minimum) points within a specific interval or neighborhood.

#### Steps to Identify Relative Extrema from a Graph:
1. **Locate Critical Points**:
- Find points where the derivative is zero or undefined (critical points).
- Critical points signal potential locations for relative extrema.

2. **Examine Neighborhoods**:
- Analyze the behavior of the function near the critical points.
- Use the first derivative test or second derivative test to determine if each critical point is a local maximum, minimum, or neither.

3. **Label Relative Extrema**:
- Mark the points where the function has local maximum or minimum values.

By scrutinizing the graph and applying calculus principles or contextual information, you can label both absolute (global) and relative (local) extrema on a function's graph.

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

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

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

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