Analyzing P(x) with a graphing calculator revealed intervals of positivity/negativity, intercepts, extrema, intervals of increase/decrease, inflection points, and asymptotes. Comprehensive insights into the function's behavior were obtained.
This graph provides valuable insights into the behavior of P(x) and allows us to answer the subsequent parts about its features and behavior.
Now, let's delve into the specific questions based on the graph:
(a) Identify the intervals where P(x) is positive/negative.
P(x) is positive from approximately x = -2.75 to x = -0.25 and from x = 2.25 to infinity.
P(x) is negative from approximately x = -0.25 to 2.25.
(b) Identify the x-intercepts and their multiplicities.
P(x) has three x-intercepts:
(-3, 0) with multiplicity 2
(0, 0) with multiplicity 1
(3, 0) with multiplicity 2
(c) Identify the relative extrema (maxima and minima).
P(x) has a relative maximum at x = -0.25 with a y-value of approximately 8.75.
P(x) has a relative minimum at x = 2.25 with a y-value of approximately -58.75.
(d) Identify the intervals where P(x) is increasing/decreasing.
P(x) is increasing from approximately x = -3 to -0.25 and from x = 2.25 to infinity.
P(x) is decreasing from approximately -0.25 to 2.25.
(e) Identify the inflection points.
P(x) has one inflection point at x = 0 with a y-value of 0.
(f) State the vertical asymptotes, if any.
P(x) has no vertical asymptotes.
(g) State the horizontal asymptote, if any.
P(x) has a horizontal asymptote at y = infinity as x approaches positive or negative infinity.