Graphing Piecewise Functions: A Comprehensive Guide

Algebra 2 Grades High School 5:09 Video
Print Lesson Plan Watch Video

Lesson Description

Master the art of graphing piecewise functions with this step-by-step lesson, covering function notation, domain restrictions, and various graphing techniques. This lesson reinforces sketching graphs based on restrictions. Aligned with Algebra 2 standards.

Video Resource

Piecewise Functions Graphing

Mario's Math Tutoring

Duration: 5:09
Watch on YouTube

Key Concepts

  • Function Notation (f(x) as y)
  • Domain Restrictions
  • Graphing Linear and Quadratic Equations
  • Open vs. Closed Intervals
  • Vertical Line Test

Learning Objectives

  • Students will be able to interpret function notation in the context of piecewise functions.
  • Students will be able to identify and apply domain restrictions when graphing piecewise functions.
  • Students will be able to graph piecewise functions using both traditional graphing methods and tables of values.
  • Students will be able to distinguish between open and closed intervals when graphing piecewise functions.
  • Students will be able to determine if a piecewise function is a function using the vertical line test.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of functions and function notation (f(x) = y). Briefly introduce piecewise functions as functions defined by multiple sub-functions, each applying to a certain interval of the domain.
  • Example 1: Graphing a Piecewise Function with Two Linear Equations (15 mins)
    Follow Mario's Math Tutoring's first example (0:20-2:44). Demonstrate how to graph each linear equation separately, then apply the domain restrictions to keep only the relevant part of each graph. Emphasize the importance of open and closed circles at the boundaries of the intervals. Also, show the alternative method of creating a table of values for each piece.
  • Example 2: Graphing a Piecewise Function with Three Equations (15 mins)
    Follow Mario's Math Tutoring's second example (2:44-end). Graph a piecewise function containing a quadratic, a constant, and a linear equation. Reinforce the concept of applying domain restrictions and using open/closed circles correctly. Highlight how to erase the parts of the graph that don't fit the domain restrictions.
  • Vertical Line Test and Function Identification (5 mins)
    Explain and demonstrate the vertical line test to ensure students understand whether the graphed piecewise function is actually a function. Review that the vertical line test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.
  • Practice Problems (10 mins)
    Provide students with practice problems to graph piecewise functions. Encourage them to use both graphing methods (traditional and table) to reinforce their understanding.

Interactive Exercises

  • Desmos Graphing Challenge
    Students use Desmos to graph piecewise functions with varying levels of difficulty. Provide specific piecewise functions with domain restrictions for students to graph, adjusting the complexity as they progress.
  • Error Analysis
    Present students with incorrectly graphed piecewise functions and ask them to identify and correct the errors. This can focus on incorrect application of domain restrictions, incorrect open/closed circles, or mis-graphed equations.

Discussion Questions

  • How does changing the domain restrictions affect the graph of a piecewise function?
  • Why is it important to use open and closed circles when graphing piecewise functions?
  • Can a piecewise function be continuous? What would make a piecewise function continuous?
  • Can all equations create a piecewise function?

Skills Developed

  • Graphing functions
  • Applying domain restrictions
  • Interpreting function notation
  • Problem-solving
  • Analytical thinking

Multiple Choice Questions

Question 1:

In a piecewise function, what does the domain restriction tell you?

Correct Answer: The interval over which a specific equation applies

Question 2:

What does an open circle on the graph of a piecewise function indicate?

Correct Answer: The endpoint is not included in the interval

Question 3:

Which of the following is the most accurate way to describe f(x) or g(x)?

Correct Answer: It's the y-value

Question 4:

What does the vertical line test determine?

Correct Answer: Whether the graph represents a function

Question 5:

When graphing a piecewise function, how do you know which parts of each equation to keep?

Correct Answer: Based on the domain restrictions

Question 6:

What is the first step in graphing a piecewise function?

Correct Answer: Graph each function separately

Question 7:

Which is not an important key concept in creating piecewise functions?

Correct Answer: Asymptotes

Question 8:

Why would you pick values of x to create a table of values?

Correct Answer: To find y coordinates on the graph

Question 9:

When creating a piecewise function including the equation y=2, what type of line would be present?

Correct Answer: Horizontal Line

Question 10:

What is the significance of using solid dots/closed circles?

Correct Answer: Values are including the point

Fill in the Blank Questions

Question 1:

The notation f(x) is essentially another way of representing the ______ value.

Correct Answer: y

Question 2:

The _________ tells you over what part of the x-axis each piece of the function is graphed.

Correct Answer: domain restriction

Question 3:

When graphing, you can use a __________ to help find ordered pairs that fit the equation.

Correct Answer: table

Question 4:

A _______ circle indicates that a point on the graph is not included in the function.

Correct Answer: open

Question 5:

The _________ test is used to determine if a graph represents a function.

Correct Answer: vertical line

Question 6:

If the domain restriction states x > 2, we know the point at x=2 should be a _______ circle.

Correct Answer: open

Question 7:

When graphing the pieces for piecewise functions, begin as though the piece goes on ________.

Correct Answer: forever

Question 8:

When considering the domain restriction, look at the ____ values.

Correct Answer: x

Question 9:

When using a _______, plug x values into the function to identify ordered pairs.

Correct Answer: table

Question 10:

Vertical Line Test states that a graph represents a function if no vertical line __________ the graph at more than one point.

Correct Answer: intersects

Educational Standards

A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically. A2.F.1.5:Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans