Piecewise Functions: Graphing Techniques and Applications

PreAlgebra Grades High School 5:18 Video
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Lesson Description

Master the art of graphing piecewise functions using two distinct methods: the direct graphing-and-erasing technique and the table-based approach. Explore domain restrictions and their impact on the graphical representation.

Video Resource

Graphing Piecewise Functions - 2 Methods

Mario's Math Tutoring

Duration: 5:18
Watch on YouTube

Key Concepts

  • Piecewise Functions
  • Domain Restrictions
  • Graphing Techniques (Direct & Table-Based)
  • Open and Closed Intervals

Learning Objectives

  • Students will be able to graph piecewise functions using the direct graphing-and-erasing method.
  • Students will be able to graph piecewise functions using the table-based method.
  • Students will be able to identify and interpret domain restrictions in the context of piecewise functions.
  • Students will be able to determine whether endpoints are included (closed circle) or excluded (open circle) based on the domain restriction.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a piecewise function. Briefly discuss why piecewise functions are useful (e.g., modeling situations with different rules over different intervals). Introduce the two graphing methods that will be covered in the video.
  • Method 1: Graphing and Erasing (15 mins)
    Play the first part of the video (Example 1). Pause at key points to explain Mario's steps. Emphasize the importance of drawing the dashed lines at the boundaries of the domain restrictions. Clearly demonstrate how to erase the unwanted portions of the graph, leaving only the part that satisfies the domain restriction. Explain the difference between open and closed circles at the endpoints and how they relate to the inequality symbols (≤, <, ≥, >).
  • Method 2: Table-Based Approach (15 mins)
    Play the second part of the video (Example 2). Explain the table method. Stress the importance of selecting appropriate x-values within each domain interval. Show how to plot the points from the table and connect them to form the graph. Again, emphasize the significance of open and closed circles at the endpoints. Highlight that this method can be useful for functions that are more difficult to graph directly.
  • Comparison and Practice (10 mins)
    Discuss the pros and cons of each method. When might one method be preferred over the other? Work through an additional example together, allowing students to choose which method they want to use. Provide immediate feedback.
  • Further Practice (5 mins)
    Assign additional practice problems that test the students knowledge of the material. Wrap up the lesson by having students ask any final questions they have. Remind them that math is fun and they can do it!

Interactive Exercises

  • Graphing Challenge
    Provide students with a set of piecewise functions with varying degrees of complexity. Have them graph each function using their preferred method. Encourage collaboration and peer review.
  • Domain Detective
    Present students with graphs of piecewise functions without the equations. Have them determine the domain restrictions for each segment of the function.

Discussion Questions

  • What are the advantages and disadvantages of each graphing method?
  • How does the domain restriction affect the graph of a piecewise function?
  • How can you determine whether to use an open or closed circle at the endpoint of a function segment?
  • How can Piecewise functions be used in the real world?

Skills Developed

  • Graphing Functions
  • Interpreting Domain Restrictions
  • Problem-Solving
  • Analytical Thinking

Multiple Choice Questions

Question 1:

What is a piecewise function?

Correct Answer: A function defined by multiple sub-functions on different intervals

Question 2:

In the direct graphing method, what is the purpose of the dashed lines?

Correct Answer: To mark the boundaries of the domain restrictions

Question 3:

What does an open circle at the endpoint of a function segment indicate?

Correct Answer: The endpoint is excluded from the domain

Question 4:

What is the primary advantage of the table-based method for graphing piecewise functions?

Correct Answer: It is easier for complex functions

Question 5:

Which inequality symbol indicates that the endpoint is included in the interval?

Correct Answer: ≤

Question 6:

When using the graphing and erasing method, what should you do after graphing the function, but before erasing?

Correct Answer: Identify which part of the graph satisfies the functions domain restriction

Question 7:

For a piecewise function, f(x) = {x+1, x<0; x^2, x>=0}, what is the value of f(-1)?

Correct Answer: 0

Question 8:

What does the term 'domain restriction' refer to in the context of piecewise functions?

Correct Answer: Specific x-values for which certain function segments are defined

Question 9:

Which of the following functions could NOT be part of a piecewise function?

Correct Answer: All can be included

Question 10:

For the piecewise function f(x) = {2x, x<=2; 5, x>2}, what is the y-value when x=2?

Correct Answer: 4

Fill in the Blank Questions

Question 1:

A function defined by multiple sub-functions, each applying to a certain interval of the domain, is called a __________ function.

Correct Answer: piecewise

Question 2:

The values of x for which a particular sub-function is defined are called the __________ __________.

Correct Answer: domain restriction

Question 3:

A __________ circle on a graph indicates that the endpoint is included in the interval.

Correct Answer: closed

Question 4:

When graphing a piecewise function, the __________ __________ method involves graphing the entire function and then erasing the unwanted portion.

Correct Answer: direct graphing

Question 5:

The __________-__________ approach to graphing piecewise functions involves creating tables of x and y values for each interval.

Correct Answer: table based

Question 6:

A(n) __________ circle indicates the endpoint is not included in the domain.

Correct Answer: open

Question 7:

The two methods for graphing a piecewise function are the graphing and erasing method, and the __________ method.

Correct Answer: table

Question 8:

Piecewise functions are useful because they allow us to model situations with __________ rules.

Correct Answer: different

Question 9:

Open and closed circles can only appear at a functions __________, when the intervals change.

Correct Answer: endpoints

Question 10:

When graphing a piece-wise function, make sure to carefully evaluate the __________ restrictions for each sub-function.

Correct Answer: domain

Educational Standards

PC.F.1.2:[Objective] Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.1.4:[Objective] Describe end behavior, asymptotic behavior, and points of discontinuity. PC.F.1.1:[Objective] Interpret characteristics of a function defined by an expression in the context of the situation.

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