Write the Word or Phrase That Best Completes Each Statement $$f ( x ) = - x ^ { 2 } ( x - 1 ) ( x + 3 )$$

question 212

Essay

Write the word or phrase that best completes each statement or answers the question.
Analyze the graph of the given function f as follows:
(a) Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.
(b) Find the x- and y-intercepts of the graph.
(c) Determine whether the graph crosses or touches the x-axis at each x-intercept.
(d) Graph f using a graphing utility.
(e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places.
(f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points.
(g) Find the domain of f. Use the graph to find the range of f.
(h) Use the graph to determine where f is increasing and where f is decreasing.
- $$f ( x ) = - x ^ { 2 } ( x - 1 ) ( x + 3 )$$

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Definitions:

Adaptive Imperative

The inherent need for organisms or systems to adapt in order to survive and thrive in their environments.

Behavioral Organism

Refers to an entity or being, typically in a psychological context, considered in terms of its behavioral responses and interactions with its environment.

Action System

A conceptual framework in sociology that views social systems as composed of interrelated actions of individuals and groups.

Systemic Evolution

Describes the process of change and development within a system, particularly in the context of complex systems and societies.