Math Problem Statement

Determine the intervals of increase and decrease for the function graphed above.

Solution

From the graph you provided, the function's intervals of increase and decrease can be determined by observing where the function rises and falls as xx increases.

Increasing Intervals

The function is increasing in the intervals where the graph moves upwards as xx progresses from left to right. From the graph:

  • The function increases from x=3x = -3 to x=1x = -1
  • The function increases again from x=1x = 1 to x=4x = 4

Decreasing Intervals

The function is decreasing in the intervals where the graph moves downwards as xx progresses from left to right. From the graph:

  • The function decreases from x=5x = -5 to x=3x = -3
  • The function decreases again from x=1x = -1 to x=1x = 1
  • The function decreases further from x=4x = 4 to x=5x = 5

Thus, the function is:

  • Increasing on the intervals: (3,1)(-3, -1) and (1,4)(1, 4)
  • Decreasing on the intervals: (5,3)(-5, -3), (1,1)(-1, 1), and (4,5)(4, 5)

Would you like to delve into how these intervals are calculated or have any other questions?

Here are some related questions you might find interesting:

  1. How do you find the exact turning points of this function?
  2. What might the algebraic expression of this function look like?
  3. How would the increasing and decreasing intervals change if the graph were shifted upwards or downwards?
  4. Can you explain the concept of concavity and points of inflection as they relate to this graph?
  5. How can derivatives help predict the behavior of a function based on its graph?

Tip: When analyzing graphs for increasing and decreasing intervals, identifying local maxima and minima is crucial as these points mark the transitions between increasing and decreasing behaviors.