Question 17·Medium·Two-Variable Data: Models and Scatterplots
A chemist measures the mass of a radioactive sample after each half-life cycle. The results are shown in the table.
| Number of half-life cycles | Mass (grams) |
|---|---|
| 0 | 200 |
| 1 | 100 |
| 2 | 50 |
| 3 | 25 |
| 4 | 12.5 |
Which type of function best models the relationship between the number of half-life cycles and the mass of the sample?
For questions asking which type of function fits a table or scatterplot, first decide if the data are increasing or decreasing as x increases. Next, quickly check for constant differences (linear) versus constant ratios (exponential) between successive y-values. If the same amount is added or subtracted each time, choose a linear model; if the same factor is multiplied each time (especially for contexts like half-life or repeated percentage change), choose an exponential model and match the direction (increasing or decreasing).
Hints
Look at how the mass changes each step
Compare each mass to the one before it. Ask: what happens when the number of half-life cycles increases by 1? Does the mass change by the same number of grams, or by the same fraction?
Decide on increasing vs. decreasing
As you go down the table (more half-life cycles), do the mass values go up or down overall? This tells you whether the function is increasing or decreasing.
Distinguish linear from exponential
For a linear relationship, the outputs change by a constant difference (like always minus 10). For an exponential relationship, the outputs change by a constant ratio (like always half as much). Check which one matches this table.
Desmos Guide
Enter the data as a table
In Desmos, create a table and enter the number of half-life cycles in the first column (x1): 0, 1, 2, 3, 4. Enter the corresponding mass values in the second column (y1): 200, 100, 50, 25, 12.5. Look at the plotted points to see whether they rise or fall as x increases.
Test a linear model
In the expression line, type y1 ~ m x1 + b to perform a linear regression. Check how well the straight line matches the points: see if all the points lie on (or very close to) the line, and note whether the line captures the pattern of the decreases.
Test an exponential model
In a new line, type y1 ~ a * b^(x1) to fit an exponential model. Compare how closely this curve passes through the data points versus the line. Also notice whether the curve goes downward or upward as x increases; that tells you the direction of change in the exponential model.
Step-by-step Explanation
Read the pattern in the table
List the masses in order of cycles: 200, 100, 50, 25, 12.5. Notice what happens each time the number of half-life cycles increases by 1.
Decide if the function is increasing or decreasing
As the number of half-life cycles increases from 0 to 4, the mass goes from 200 down to 12.5. Because the outputs (mass) get smaller as the inputs (cycles) get larger, the relationship is decreasing, not increasing.
Compare linear and exponential patterns
For a linear relationship, the -values change by the same difference each time (add or subtract a constant). Here the differences are:
- These differences are not equal, so it is not linear.
For an exponential relationship, the -values change by the same factor (ratio) each time (multiply by a constant). Here the ratios are:
- The constant ratio of shows the pattern is exponential.
Combine the direction with the type of pattern
You have found that:
- The function is decreasing (mass goes down as cycles go up), and
- The pattern is exponential (multiply by the same factor each time).
Therefore, the relationship is best modeled by a decreasing exponential function.