Question 10·Easy·Two-Variable Data: Models and Scatterplots
A biologist observes a colony of bacteria that starts with 120 cells and the population doubles every 30 minutes. What type of function best models the relationship between the number of bacteria in the colony and time?
For "type of function" word problems, first check whether the quantity changes by a constant amount (add/subtract the same number each time) or by a constant factor/percentage (multiply by the same number, like doubling or halving). Constant amounts indicate a linear model; constant factors indicate an exponential model. Then decide whether the quantity is going up or down over time to choose between increasing or decreasing. This quick two-step check lets you match situations to linear vs. exponential, increasing vs. decreasing, efficiently on the SAT.
Hints
Look at the key phrase
Pay close attention to the words "doubles every 30 minutes". What does "doubles" tell you about how the number changes?
Think about how the change happens
Decide whether the bacteria count is changing by adding the same amount each time, or by multiplying by the same factor each time.
Determine the direction of change
Is the number of bacteria getting larger or smaller over time? Use that to decide between "increasing" and "decreasing" in the answer choices.
Desmos Guide
Enter an exponential model
In Desmos, type an equation that matches the description, such as y = 120 * 2^x, where x represents the number of 30-minute intervals.
Observe the shape and direction
Look at how the graph behaves as x increases: notice whether it curves upward and how quickly it grows compared with a straight line.
Compare with a linear model (optional)
Add a linear function like y = 120 + 50x and compare its straight-line growth to the curved, faster-growing graph of the doubling model to see why a linear description does not fit.
Step-by-step Explanation
Identify the type of change
Focus on the phrase "doubles every 30 minutes".
- "Doubles" means the number of bacteria is multiplied by 2.
- This happens in equal time intervals (every 30 minutes).
When a quantity changes by being multiplied by the same factor in each equal time step, that pattern is exponential, not linear.
Decide if the function is increasing or decreasing
Ask: Is the number of bacteria going up or down over time?
- Starting from 120 cells, the population doubles, so it becomes 240, then 480, and so on.
- The population is clearly growing, not shrinking.
So the function that models this situation must be increasing, not decreasing.
Match the pattern to the answer choice
You have two important facts:
- The change is multiplicative (doubling) over equal time intervals - this is exponential.
- The population is growing over time - this is increasing.
Therefore, the type of function that best models the relationship is increasing exponential.