Question

scientists recorded the population of beetles in an area during a 29 - month period. they plotted the data from the study, and they used a quartic regression to generate the polynomial function that best represents the data.
(a) during what month(s) of the study were there the most beetles, and approximately how many beetles were there?
(b) during what month(s) of the study were the number of beetles predicted by the function unreasonable? explain your reasoning.
(c) during which months were the number of beetles increasing?

Explanation:

Step1: Analyze the graph for maximum population

Look for the highest point(s) on the graph. The \(y\) - value represents the number of beetles and the \(x\) - value represents the month.

Step2: Identify unreasonable predictions

Check for parts of the function (not the data points) that don't make sense. A population can't be negative.

Step3: Determine increasing intervals

Look at the graph where the function is going up (from left - to - right).

Answer:

a. The most beetles were during month \(20\) and there were approximately \(10000\) beetles.
b. The function predicts a negative population (which is impossible) for \(x < 5\) (before month \(5\)). Since the number of beetles (a population) cannot be negative, the predictions for \(x < 5\) are unreasonable.
c. The number of beetles was increasing during the months \(5 < x<20\) (from month \(5\) to month \(20\)).