Math Problem Statement

The function graphed above is increasing on the interval(s): and decreasing on the interval(s):

Solution

To determine the intervals where the function is increasing or decreasing, we need to observe where the graph goes upwards or downwards as we move from left to right.

Increasing Intervals:

The function is increasing when the graph moves upwards. From the graph:

  1. The function increases from approximately (4,2)(-4, 2) to (2,0)(-2, 0).
  2. It also increases from approximately (0,2)(0, -2) to (2,2)(2, 2).

So, the increasing intervals are:

  • (4,2)(-4, -2)
  • (0,2)(0, 2)

Decreasing Intervals:

The function is decreasing when the graph moves downwards. From the graph:

  1. The function decreases from approximately (,4)(-∞, -4).
  2. It also decreases from approximately (2,0)(-2, 0) to (0,2)(0, -2).
  3. Finally, it decreases from (2,2)(2, 2) to (,)(∞, ∞).

So, the decreasing intervals are:

  • (,4)(-∞, -4)
  • (2,0)(-2, 0)
  • (2,)(2, ∞)

Would you like to see more details or have any further questions?

Here are five related questions that might help expand on this topic:

  1. How can you find intervals of increase and decrease using derivatives?
  2. What are the critical points of a function, and how are they related to intervals of increase and decrease?
  3. How do you determine concavity and inflection points from a graph?
  4. What role do maxima and minima play in determining increasing and decreasing intervals?
  5. How can you use the first derivative test to determine the nature of intervals?

Tip: To confirm the intervals where a function is increasing or decreasing, you can use the first derivative test.

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Math Problem Analysis

Mathematical Concepts

Graphing Functions
Intervals of Increase and Decrease

Formulas

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Theorems

First Derivative Test
Monotonicity

Suitable Grade Level

Grades 10-12