Math Problem Statement

Use the graph to determine: a. open intervals on which the function is increasing, if any. b. open intervals on which the function is decreasing, if any. c. open intervals on which the function is constant, if any.

Solution

Let's analyze the graph and answer the questions:

Observations from the graph:

  1. The graph shows the function as:
    • Decreasing from x=4x = -4 to x=0x = 0.
    • Constant from x=0x = 0 to x=6x = 6.

Answers:

a. Open intervals on which the function is increasing:

  • There is no interval where the function is increasing.
  • Answer: No interval of increase\text{No interval of increase}.

b. Open intervals on which the function is decreasing:

  • The function decreases on the interval (4,0)(-4, 0).
  • Answer: (4,0)(-4, 0).

c. Open intervals on which the function is constant:

  • The function is constant on the interval (0,6)(0, 6).
  • Answer: (0,6)(0, 6).

Would you like further clarification or a detailed explanation of interval notation?


Questions to expand your understanding:

  1. How do you determine where a function is increasing or decreasing from a graph?
  2. What is the significance of open intervals in describing functions?
  3. Can a function have overlapping intervals of increase and decrease?
  4. How do you interpret constant intervals on a graph?
  5. What role do endpoints play in identifying intervals?

Tip: Always check the slope of the function visually—positive slope means increasing, negative slope means decreasing, and zero slope means constant.

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Math Problem Analysis

Mathematical Concepts

Functions
Increasing and Decreasing Intervals
Graph Analysis

Formulas

Interval notation

Theorems

Behavior of functions based on their slopes

Suitable Grade Level

Grades 9-11