The scatterplot shows the estimated population of a species of insect at various numbers of days after hatching. Each dot represents one observation. A curve representing an exponential model is shown. The two coordinate labels on the curve indicate exact values of the model.
If is the population, in thousands, predicted by the model days after hatching and , what is the value of ?
For an exponential model, focus on multiplicative change rather than additive change. First use two clearly labeled points to determine the multiplier over a convenient repeated interval. Then count how many of those intervals occur between the two input values in the question and raise the multiplier to that number of intervals.
Hints
Compare the labeled points
Use the two labeled points on the model curve to find how the predicted population changes from day 2 to day 8.
Use equal time intervals
The change from day 2 to day 8 occurs over 6 days. Think about dividing those 6 days into equal 2-day intervals.
Match the target interval
The interval from day 4 to day 14 contains five 2-day intervals. Apply the same exponential multiplier once for each interval.
Desmos Guide
Use the labeled points to identify the pattern
Algebra is fastest here. The labeled values show that the model changes from to over 6 days, so it has a factor of over three 2-day intervals. Thus, the model doubles every 2 days.
Enter an equivalent model
Enter f(x)=6*2^((x-2)/2). This model has and doubles whenever increases by 2.
Compare the two predictions
Enter f(14)/f(4). The resulting value is the multiplier .
Step-by-step Explanation
Use the labeled model values
The curve passes through and . Therefore,
The population is multiplied by from day 2 to day 8.
Find the repeated time interval
The interval from day 2 to day 8 is 6 days, which contains three 2-day intervals. If the population is multiplied by every 2 days, then
So : the model predicts that the population doubles every 2 days.
Apply the multiplier to the requested interval
From day 4 to day 14 is 10 days, or five 2-day intervals. Thus,
Therefore, .