PAP Algebra II- Piecewise

Unraveling Piecewise Functions

Understanding the mystery behind piecewise functions and their applications in mathematics.

Introduction to Piecewise Functions

• A piecewise function is a function that is defined by multiple sub-functions, each applying to a different interval of the input domain.

• It is represented using different equations for different intervals.

• Each sub-function is valid only for its specific interval.

• Piecewise functions are commonly used to model real-world situations with changing conditions.

Multiple Choice

What is a piecewise function?

• A function that is defined by multiple sub-functions

• A function that is valid only for a specific interval

• A function that models real-world situations with changing conditions

• A function that represents different equations for different intervals

Piecewise Functions

A piecewise function is defined by multiple sub-functions. It represents different equations for different intervals. Piecewise functions are used to model real-world situations with changing conditions. They are valid only for specific intervals, making them a powerful tool in mathematics and physics.

Example Piecewise Functions

A moving company charges according to the weight of boxes packed. If the box's weight is 10 pounds and under, the company charges $20. If the box weighs between 10 and up to 50 pounds, the company charges $35. If the weight is over 50 pounds, the company charges $40, plus an additional $2 for each pound.

Multiple Choice

What are piecewise functions used for?

• Modeling real-world situations with changing conditions

• Solving complex mathematical equations

• Representing linear functions

• Graphing exponential functions

Piecewise Functions:

Trivia: Every income-earning person in the US uses an application of piecewise functions at least once per year.

Evaluating Piecewise Functions

• Identify the different pieces of a piecewise function

• Evaluate each piece of the function separately

• Compare the sub-functions to form an idea of what is happening over the entire domain

Multiple Choice

What is the purpose of evaluating each piece of a piecewise function separately?

• To understand what one interval represents

• To identify the different types of functions

• To understand how the range is affected by each interval of the domain

• To unravel the mysteries of the universe

Applications of Piecewise Functions

• Modeling tiered price structure

• Optimizing resource allocation

• Designing efficient algorithms

• Representing complex systems

Multiple Select

Which are application of piecewise functions?

• Modeling tiered price structures

• Optimizing resource allocation

• Designing efficient algorithms

• Representing the weather

Graphing Piecewise Functions

1. Identify the different intervals or domains of the function. 2. Determine the expression for each interval. 3. Graph each expression on its respective interval. 4. Use open or closed circles to indicate whether the endpoints are included or excluded. 5. Connect the graphs of each interval to create the piecewise function graph.

Practice

Use the provided space to graph and answer the following:

The Hewitt City Council recently approved increases in water rates. There is a base fee of $67 for the first 3,000 gallons of water used. The monthly charge for each additional 1,000 gallons is $5.35 up to 7,000 gallons and $6 between 7,000 and 12,000 gallons. The charge for use over 12,000 gallons is $6.45.

If Mr. McAdoo uses 8,540 gallons this month, what will he pay for water?