Question 28·Easy·Two-Variable Data: Models and Scatterplots
A biologist records the size of a bacterial colony every 30 minutes. Each new measurement shows the colony is about 1.5 times as large as it was 30 minutes earlier. What type of function best models the relationship between the colony’s size and the elapsed time?
For growth/decay word problems, first decide if the quantity is increasing or decreasing, then decide if the change is by addition/subtraction (linear) or multiplication by a factor (exponential). Phrases like "times as large" or "multiplied by" with a fixed time interval signal exponential models, while "more than" or "less than" by a fixed amount per interval signal linear models. Match this combination (increasing vs. decreasing, linear vs. exponential) directly to the answer choices to save time.
Hints
Direction of change
Focus on the words describing what happens to the colony each time it is measured. Is the colony getting bigger or smaller?
Type of change: add vs. multiply
Pay attention to the phrase "1.5 times as large." Does that suggest adding a fixed number each time or multiplying by a fixed number each time?
Function type for repeated multiplication
When a quantity is repeatedly multiplied by the same factor over equal time intervals, what type of function usually models that situation?
Combine your observations
Once you know whether the colony is increasing or decreasing and whether the change is additive or multiplicative, match that combination to the best function type in the answer choices.
Desmos Guide
Create a simple exponential model
In Desmos, enter an example function such as f(x) = 100*(1.5)^(x/30), where x is the time in minutes and 100 is a sample starting size.
Use a table to see the pattern
Click the gear icon next to the function and select "Table". In the x-column, enter 0, 30, 60, 90, etc. Look at how the y-values change: notice they grow faster and faster rather than by a constant difference.
Compare with a linear model
Add a linear function like g(x) = 100 + 20*(x/30) and create its table as well. Compare the two tables and graphs: one will increase by the same amount each step (a straight line), while the other will increase by multiplying by 1.5 each step (a curved growth pattern). Use this to decide which type of model best matches the situation.
Step-by-step Explanation
Decide if the colony is growing or shrinking
The problem says, "Each new measurement shows the colony is about 1.5 times as large as it was 30 minutes earlier."
That means every time you measure, the colony is bigger than before, not smaller. So the relationship between time and size is increasing, not decreasing.
Decide if the change is additive or multiplicative
"1.5 times as large" means you multiply the previous size by each time, not add a fixed amount.
For example, if the colony started at 100 bacteria:
- After 30 minutes:
- After 60 minutes:
- After 90 minutes:
Notice the amount added each time is changing (50, then 75, then 112.5), so it is not linear (linear means you add the same amount each step). Repeated multiplication by the same factor over equal time intervals is modeled by an exponential function, not a linear one.
Combine “increasing” with the function type
From Step 1, the colony size is increasing over time. From Step 2, the relationship is exponential, because we repeatedly multiply by the same factor.
Putting these together, the function that best models the relationship is an increasing exponential function, which corresponds to answer choice A) Increasing exponential.