Stepping Up to the Greatest Integer Function: Graphing & Transformations

Algebra 2 Grades High School 5:30 Video
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Lesson Description

Explore the greatest integer function (step function), its graph, and how transformations affect it. Learn to graph and analyze functions of the form y = [x], y = -[x], y = a[x], and y = [bx].

Video Resource

Greatest Integer Function How to Graph

Mario's Math Tutoring

Duration: 5:30
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Key Concepts

  • Greatest Integer Function (Step Function) definition and notation
  • Graphing the greatest integer function
  • Transformations of the greatest integer function: reflection, vertical stretch/shrink, horizontal stretch/shrink

Learning Objectives

  • Students will be able to define and use the notation for the greatest integer function.
  • Students will be able to graph the basic greatest integer function (y = [x]).
  • Students will be able to identify and graph transformations of the greatest integer function, including reflections, vertical stretches/shrinks, and horizontal stretches/shrinks.

Educator Instructions

  • Introduction (5 mins)
    Begin by introducing the greatest integer function and its notation. Explain the concept of rounding down to the nearest integer. Show examples using a number line to illustrate the rounding process.
  • Graphing the Basic Greatest Integer Function (15 mins)
    Demonstrate how to create a table of values for the greatest integer function. Emphasize the importance of including both integer and non-integer values. Guide students through plotting the points and drawing the step-like graph. Highlight the open and closed endpoints of each step.
  • Transformations of the Greatest Integer Function (20 mins)
    Explain how different transformations affect the graph of the greatest integer function. Cover reflections over the x-axis (y = -[x]), vertical stretches/shrinks (y = a[x]), and horizontal stretches/shrinks (y = [bx]). Provide examples and guide students through graphing each type of transformation. Ask them to predict the change before you demonstrate.
  • Practice and Application (10 mins)
    Provide students with practice problems involving graphing the greatest integer function and its transformations. Encourage them to use the table of values method and to think about how the transformations affect the shape and position of the graph.

Interactive Exercises

  • Graphing Challenge
    Give students a worksheet with different greatest integer functions and transformations. They must graph each function accurately, labeling key points and asymptotes, if any.
  • Transformation Match
    Provide students with a list of greatest integer functions and a list of corresponding transformations (e.g., reflection over the x-axis, vertical stretch by a factor of 2). Students must match each function with its correct transformation.

Discussion Questions

  • How does the greatest integer function relate to rounding down?
  • What are the key features of the graph of the greatest integer function?
  • How do different transformations change the graph of the greatest integer function?
  • Can you think of any real-world situations that can be modeled using the greatest integer function?

Skills Developed

  • Graphing functions
  • Understanding transformations of functions
  • Applying the concept of rounding down
  • Analytical thinking

Multiple Choice Questions

Question 1:

What is the value of the greatest integer function for x = -2.3?

Correct Answer: -3

Question 2:

The graph of the greatest integer function is best described as:

Correct Answer: A series of steps

Question 3:

What transformation does y = -[x] represent compared to y = [x]?

Correct Answer: Reflection over the x-axis

Question 4:

What transformation does y = 3[x] represent compared to y = [x]?

Correct Answer: Vertical stretch by a factor of 3

Question 5:

What transformation does y = [2x] represent compared to y = [x]?

Correct Answer: Horizontal shrink by a factor of 2

Question 6:

On the graph of y=[x], each step has a:

Correct Answer: Closed circle on the left, open circle on the right

Question 7:

What is the domain of the parent greatest integer function, y=[x]?

Correct Answer: All real numbers

Question 8:

What is the range of the parent greatest integer function, y=[x]?

Correct Answer: All integers

Question 9:

The greatest integer function is also known as the:

Correct Answer: Step function

Question 10:

Which transformation will make the 'steps' of the greatest integer function shorter?

Correct Answer: y = [2x]

Fill in the Blank Questions

Question 1:

The notation for the greatest integer function is f(x) = ____.

Correct Answer: [x]

Question 2:

The greatest integer function always rounds down to the nearest ____.

Correct Answer: integer

Question 3:

Reflecting the graph of y=[x] over the x-axis results in the function y = ____.

Correct Answer: -[x]

Question 4:

Multiplying the greatest integer function by a constant outside the brackets results in a _____ stretch or shrink.

Correct Answer: vertical

Question 5:

Multiplying the x-value inside the greatest integer function brackets by a constant results in a _____ stretch or shrink.

Correct Answer: horizontal

Question 6:

On the graph of the parent greatest integer function, y=[x], the left endpoint of each step is a _____ circle.

Correct Answer: closed

Question 7:

On the graph of the parent greatest integer function, y=[x], the right endpoint of each step is a _____ circle.

Correct Answer: open

Question 8:

The value of [5.99] is _____.

Correct Answer: 5

Question 9:

The value of [-3.2] is _____.

Correct Answer: -4

Question 10:

The 'steps' of a transformed greatest integer function, y=[4x], are _____ than the steps of y=[x].

Correct Answer: shorter

Educational Standards

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.1.1: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.

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