Unlocking Function Behavior: Increasing, Decreasing, and Constant Intervals

PreAlgebra Grades High School 3:50 Video
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Lesson Description

Explore how to identify and express intervals where functions increase, decrease, or remain constant using interval notation. Learn to interpret graphs and apply proper notation conventions.

Video Resource

Interval notation for where functions Increase, Decrease, Constant

Mario's Math Tutoring

Duration: 3:50
Watch on YouTube

Key Concepts

  • Increasing, Decreasing, and Constant Intervals
  • Interval Notation (Parentheses vs. Brackets)
  • X-Values as the Domain of Analysis
  • Graphical Interpretation of Function Behavior

Learning Objectives

  • Students will be able to identify intervals where a function is increasing, decreasing, or constant from its graph.
  • Students will be able to express these intervals using correct interval notation.
  • Students will be able to differentiate between open and closed intervals when describing function behavior.
  • Students will be able to explain why x-values, not y-values, are used to define these intervals.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concept of functions and their graphical representation. Introduce the idea of function behavior: increasing, decreasing, and constant. Explain that this lesson will focus on identifying and describing these behaviors using interval notation.
  • Video Viewing (10 mins)
    Play the 'Interval notation for where functions Increase, Decrease, Constant' video by Mario's Math Tutoring. Encourage students to take notes on key concepts and examples.
  • Concept Discussion (10 mins)
    Discuss the key concepts presented in the video: * Defining Increasing, Decreasing, and Constant: What does it mean for a function to be increasing, decreasing, or constant on an interval? * X-Values vs. Y-Values: Emphasize that we use x-values to define the intervals, not y-values. * Interval Notation: Review the difference between parentheses (open interval, excluding endpoints) and brackets (closed interval, including endpoints). Explain why we use parentheses for increasing, decreasing, and constant intervals (endpoint is a transition point).
  • Guided Practice (15 mins)
    Present several graphs of functions (linear, quadratic, cubic) on the board or using a projector. For each graph, guide students through the process of identifying the intervals where the function is increasing, decreasing, and constant. Write the intervals using correct interval notation. Emphasize the use of parentheses.
  • Independent Practice (15 mins)
    Provide students with a worksheet containing several more graphs of functions. Have them independently identify and express the intervals where each function is increasing, decreasing, and constant. Circulate to provide assistance as needed.
  • Wrap-up and Assessment (5 mins)
    Review the key concepts and answer any remaining questions. Briefly introduce the multiple-choice quiz and fill-in-the-blank quiz that will assess their understanding.

Interactive Exercises

  • Graph Sketching
    Provide students with descriptions of function behavior (e.g., increasing from negative infinity to -2, constant from -2 to 1, increasing from 1 to infinity). Have them sketch a graph that matches the description.
  • Online Graphing Tool
    Use an online graphing tool like Desmos or GeoGebra to display a function. Have students collaboratively identify the increasing, decreasing, and constant intervals and write them on the board.

Discussion Questions

  • Why do we use parentheses instead of brackets when writing intervals of increasing, decreasing, or constant behavior?
  • How would you explain to someone the difference between using x-values and y-values when determining these intervals?
  • Can a function be both increasing and decreasing at the same point? Why or why not?

Skills Developed

  • Graphical Interpretation
  • Analytical Thinking
  • Mathematical Notation
  • Problem-Solving

Multiple Choice Questions

Question 1:

Which value is used to define the interval of increasing or decreasing?

Correct Answer: X-value

Question 2:

What type of interval is used when a function is increasing or decreasing?

Correct Answer: Open interval

Question 3:

The function is increasing from (-∞, -3). What does this mean?

Correct Answer: As x increases, y increases

Question 4:

In interval notation, what does a parenthesis indicate?

Correct Answer: The endpoint is excluded

Question 5:

Which of the following intervals represents where the given function, f(x), is decreasing: f(x) = -x^2 for x > 0

Correct Answer: (0, ∞)

Question 6:

A function is constant on the interval [2, 5]. What does this tell you about the function's graph on this interval?

Correct Answer: The graph is a horizontal line

Question 7:

What is the correct interval notation for a function that is increasing for all real numbers?

Correct Answer: (-∞, ∞)

Question 8:

A function transitions from increasing to decreasing at x = 3. How would you represent this transition point in interval notation for the increasing interval?

Correct Answer: (..., 3)

Question 9:

For which of the following function types, in its basic form, is there no increasing or decreasing interval?

Correct Answer: Linear Function

Question 10:

A function is increasing from (-∞, a) and increasing from (b, ∞). What symbol connects these two intervals in interval notation?

Correct Answer: ∪

Fill in the Blank Questions

Question 1:

We use __________ values to determine the intervals of increasing, decreasing, and constant function behavior.

Correct Answer: x

Question 2:

Interval notation uses __________ to exclude endpoints.

Correct Answer: parentheses

Question 3:

A function is __________ if its y-values increase as its x-values increase.

Correct Answer: increasing

Question 4:

If the graph of a function is a horizontal line, the function is said to be __________.

Correct Answer: constant

Question 5:

The symbol '∪' is used to indicate the __________ of two intervals.

Correct Answer: union

Question 6:

A function is said to be _________ if its y-values decrease as its x-values increase.

Correct Answer: decreasing

Question 7:

The interval __________ represents all real numbers.

Correct Answer: (-∞, ∞)

Question 8:

The point where a function changes from increasing to decreasing is called a relative __________.

Correct Answer: maximum

Question 9:

A square bracket in interval notation indicates the endpoint is __________.

Correct Answer: included

Question 10:

For a linear equation that is constantly decreasing, the slope is __________.

Correct Answer: negative

Educational Standards

PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.3:Interpret characteristics of graphs and tables for a function that models a relationship between two quantities in terms of the quantities. PC.F.1.4:Describe end behavior, asymptotic behavior, and points of discontinuity.

Teaching Materials

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