Question 20·Medium·Two-Variable Data: Models and Scatterplots
A laboratory technician measured the mass of a sample of a radioactive isotope at regular time intervals. The results are shown below.
| Time (hours) | Mass (g) |
|---|---|
| 0 | 160 |
| 1 | 80 |
| 2 | 40 |
| 3 | 20 |
| 4 | 10 |
Which type of function best models how the mass of the sample changes over time?
For questions that ask whether data are best modeled by linear or exponential functions, first decide if the values are going up or down as increases (increasing vs. decreasing). Then check the pattern: compute a few differences between consecutive -values—if they are constant, the model is linear. If not, compute ratios of consecutive -values—if those are constant, the model is exponential. This quick difference-then-ratio check lets you choose the correct model efficiently without writing full equations.
Hints
Look at the overall direction
As time increases from 0 to 4 hours, does the mass go up or go down? This tells you whether the function is increasing or decreasing.
Test for linear change
For a linear function, the difference between consecutive -values (masses) should be the same. Subtract each mass from the previous one and see if those differences match.
Test for exponential change
If the differences are not the same, try dividing each mass by the previous one. Is the ratio between consecutive masses the same each time? That suggests an exponential pattern.
Desmos Guide
Enter the data as a table
In Desmos, click the "+" button and choose Table. In the first column (x), enter the times 0, 1, 2, 3, 4. In the second column (y), enter the corresponding masses 160, 80, 40, 20, 10.
Look at the shape of the plotted points
Check the graph of the points. Ask yourself: Do the points form a straight line, or do they curve? If they do not lie on a straight line but instead bend, the relationship is not linear.
Decide how the graph changes
Notice whether the graph goes up or down as increases, and whether it bends more sharply as it goes. Use this information (direction of change and curve vs. straight line) to match the graph to the answer choice that describes both the direction (increasing or decreasing) and the type (linear or exponential).
Step-by-step Explanation
Decide if the function is increasing or decreasing
Look at how the mass changes as time increases.
- At time 0 hours, the mass is 160 g.
- At 1 hour, it is 80 g.
- At 2 hours, it is 40 g, and it keeps getting smaller.
Since the mass gets smaller as time goes on, the function is decreasing, not increasing.
Check whether the change is linear (constant difference)
For a linear function, the change in (here, mass) is the same each time (time) increases by 1.
Compute the differences between consecutive masses:
These differences are not the same, so the relationship is not linear.
Check whether the change is exponential (constant ratio)
For an exponential function, each -value is multiplied by the same factor to get the next one.
Compute the ratios of consecutive masses:
Each hour, the mass is multiplied by . This constant ratio means the data follow an exponential pattern.
Combine both ideas to choose the model
You found that:
- The mass is decreasing over time.
- The change follows a constant ratio, so it is exponential, not linear.
Therefore, the type of function that best models how the mass changes over time is decreasing exponential.