Question 37·Easy·Two-Variable Data: Models and Scatterplots
A biologist records the size of a bacterial culture every hour. The results are shown below.
| Time (hours) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Population (cells) | 200 | 400 | 800 | 1,600 | 3,200 |
What type of function best models the relationship between the bacterial population and time?
When you see a table of two-variable data and are asked what type of model fits best, quickly check two things: (1) Is the dependent variable going up or down as the independent variable increases? (2) Are the changes between -values constant differences (suggesting linear) or constant ratios (suggesting exponential)? Subtract consecutive -values to test for linear behavior; if those differences aren’t constant, divide consecutive -values to see if there is a constant factor instead. Then match “constant amount vs. constant factor” and “increasing vs. decreasing” to the best function type among the answer choices.
Hints
Focus on how the population changes
Compare each population value to the one before it. Ask: from one hour to the next, are we adding the same number, or doing something else?
Try subtracting consecutive values
Compute , , , and . Are these differences all the same, or do they change?
Now try dividing consecutive values
Compute , , , and . What do you notice about these ratios, and what type of function usually goes with a constant ratio instead of a constant difference?
Consider both direction and pattern
Think about two questions: (1) Is the population going up or down as time increases? (2) Does it change by equal steps or by equal factors?
Desmos Guide
Enter the data as a table
Type + and choose Table, then enter the time values 0, 1, 2, 3, 4 in the column and the populations 200, 400, 800, 1600, 3200 in the column. Look at how the plotted points are arranged on the graph.
Check whether a line fits the data
In a new expression line, type y1 ~ m x1 + b to perform a linear regression. Look at the graph: do the data points lie close to a straight line, or do they curve away from the line as increases?
Check whether an exponential curve fits the data
In another expression line, type y1 ~ a b^(x1) to perform an exponential regression. Compare how well this curve matches the points versus the line. Notice whether the curve captures the rapidly increasing pattern seen in the table; that will tell you which type of function from the choices best models the data.
Step-by-step Explanation
Look at how the population changes each hour
Write the populations in order of time: 200, 400, 800, 1{,}600, 3{,}200.
Now find how much the population increases each hour:
- From 0 to 1 hour:
- From 1 to 2 hours:
- From 2 to 3 hours:
- From 3 to 4 hours:
These differences are not the same; they are getting larger each hour, so the data do not increase by a constant amount.
Check for a constant ratio (multiplying pattern)
Now compare each population to the one before it by division:
Each hour, the population is multiplied by the same factor, 2. That means there is a constant ratio from one hour to the next.
Decide the function type from the pattern
A constant difference between -values corresponds to a linear function. A constant multiplicative factor (constant ratio) corresponds to an exponential function.
Here, the population is always multiplied by 2 and the values are getting larger over time, so the relationship is exponential and it is increasing, which matches answer choice A) Increasing exponential.