Unlocking Functions: Mastering Increasing, Decreasing, and Constant Intervals

PreAlgebra Grades High School 8:26 Video
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Lesson Description

Learn to identify and express intervals where a function is increasing, decreasing, or constant using interval notation. This lesson builds essential skills for analyzing function behavior and understanding calculus concepts.

Video Resource

Intervals where the Function is Increasing, Decreasing, or Constant

Mario's Math Tutoring

Duration: 8:26
Watch on YouTube

Key Concepts

  • Increasing Function: A function is increasing on an interval if its y-values increase as its x-values increase (reading from left to right).
  • Decreasing Function: A function is decreasing on an interval if its y-values decrease as its x-values increase (reading from left to right).
  • Constant Function: A function is constant on an interval if its y-values remain the same as its x-values increase (horizontal line).
  • Interval Notation: A method for writing sets of numbers using parentheses and brackets to indicate whether endpoints are included or excluded.
  • Open Intervals: Using parentheses to exclude endpoints in interval notation. Critical when describing increasing/decreasing/constant behavior at transition points.

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, excluding endpoints where the function changes behavior.
  • Students will be able to analyze function behavior around points of discontinuity and asymptotes.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definitions of increasing, decreasing, and constant functions. Emphasize the 'reading from left to right' analogy and the concept of slope (positive, negative, zero) to visualize function behavior. Introduce the importance of using interval notation to accurately describe these intervals.
  • Video Presentation (15 mins)
    Play the Mario's Math Tutoring video "Intervals where the Function is Increasing, Decreasing, or Constant." Encourage students to take notes on the examples provided. Pause the video after each example to allow for questions and clarification.
  • Guided Practice (15 mins)
    Work through additional examples as a class, similar to those in the video. Focus on identifying key features of the graph, such as turning points and asymptotes. Guide students in writing the intervals using proper notation. Reinforce the reason for excluding endpoints.
  • Independent Practice (10 mins)
    Provide students with a worksheet containing a variety of graphs. Instruct them to identify and express the intervals where each function is increasing, decreasing, or constant. Encourage peer collaboration and discussion.

Interactive Exercises

  • Graphing Activity
    Use graphing software (Desmos, GeoGebra) to allow students to manipulate functions and observe how changes in the function's equation affect the intervals of increasing, decreasing, and constant behavior. Challenge them to create functions with specific increasing/decreasing intervals.
  • Error Analysis
    Present students with examples of incorrectly written interval notations for increasing/decreasing intervals. Have them identify the error and correct it, explaining their reasoning.

Discussion Questions

  • Why do we use open intervals (parentheses) instead of closed intervals (brackets) when describing increasing, decreasing, or constant behavior at the points where the function changes direction?
  • How does the presence of a vertical asymptote affect the intervals where a function is increasing or decreasing?
  • Can a function be both increasing and decreasing at the same point? Explain your reasoning.
  • What are some real-world scenarios where understanding increasing and decreasing intervals of a function could be useful?

Skills Developed

  • Graph Analysis
  • Interval Notation
  • Critical Thinking
  • Problem Solving

Multiple Choice Questions

Question 1:

A function is said to be increasing on an interval if, as x increases, y:

Correct Answer: Increases

Question 2:

Which of the following interval notations indicates that the endpoint is NOT included in the interval?

Correct Answer: ()

Question 3:

On which interval is the function f(x) = x² decreasing?

Correct Answer: (-∞, 0)

Question 4:

A horizontal line on a graph represents a function that is:

Correct Answer: Constant

Question 5:

Why are endpoints typically excluded when writing intervals of increasing or decreasing behavior?

Correct Answer: The function is changing direction at the endpoints.

Question 6:

What does the union symbol (∪) represent when describing intervals?

Correct Answer: Joining of intervals

Question 7:

Consider a function with a vertical asymptote at x = 3. How would you express an interval approaching this asymptote?

Correct Answer: (...,3)

Question 8:

If a function is neither increasing nor decreasing on an interval, it is said to be:

Correct Answer: Constant

Question 9:

Which of the following functions is constant on the interval (-∞, ∞)?

Correct Answer: f(x) = 5

Question 10:

A function is decreasing when its derivative is:

Correct Answer: Negative

Fill in the Blank Questions

Question 1:

A function is __________ on an interval if its y-values decrease as its x-values increase.

Correct Answer: decreasing

Question 2:

__________ notation is used to express the intervals where a function is increasing, decreasing, or constant.

Correct Answer: Interval

Question 3:

Parentheses ( ) in interval notation indicate that the __________ are not included.

Correct Answer: endpoints

Question 4:

A function is __________ if its graph is a horizontal line.

Correct Answer: constant

Question 5:

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

Correct Answer: union

Question 6:

Vertical __________ can affect the intervals of increasing or decreasing behavior.

Correct Answer: asymptotes

Question 7:

When reading a graph to determine increasing/decreasing intervals, we read from __________ to __________.

Correct Answer: left, right

Question 8:

On the interval (0, ∞), f(x) = √x is a(n) __________ function.

Correct Answer: increasing

Question 9:

If f'(x) < 0 on an interval, then f(x) is __________ on that interval.

Correct Answer: decreasing

Question 10:

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

Correct Answer: maximum

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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