Ask yourself: between each measurement, does the mass go down by the same number of milligrams, or is it multiplied by the same number each time?

Linear functions involve adding or subtracting the same amount each step. Another common function type involves repeatedly multiplying by the same factor. Which option matches that second kind of pattern?

In Desmos, add a table. In the first column (say $x_1$), enter the times: 0, 5, 10, 15, 20. In the second column ($y_1$), enter the masses: 160, 80, 40, 20, 10. Look at how the plotted points fall on the coordinate plane.

Below the table, type y1 ~ m x1 + b to create a linear regression. Check whether this line passes close to all of your data points or if some points are clearly off the line. This tells you how well a linear model fits the data.

Now try a different type of model, such as y1 ~ a b^(x1) (an exponential form) or y1 ~ a x1^2 + b x1 + c (a quadratic form). Compare how closely each curve fits the five data points. Notice which type of model goes through all the points in a way that matches the halving pattern of the masses; that is the function type you should choose in the problem.

Write down the masses in order of time:

• At 0 hours: 160 mg
• At 5 hours: 80 mg
• At 10 hours: 40 mg
• At 15 hours: 20 mg
• At 20 hours: 10 mg

Now compare each value to the previous one:

• From 160 to 80
• From 80 to 40
• From 40 to 20
• From 20 to 10

Notice that each value is half of the previous one.

First, look at the differences between consecutive masses:

• $80 - 160 = -80$
• $40 - 80 = -40$
• $20 - 40 = -20$
• $10 - 20 = -10$

The differences are $-80, -40, -20, -10$. These are not constant.

Now look at the ratios (new value divided by previous value):

• $80/160 = 1/2$
• $40/80 = 1/2$
• $20/40 = 1/2$
• $10/20 = 1/2$

The ratio is always $1/2$, so the mass is repeatedly multiplied by the same factor, $1/2$, every 5 hours.

Recall the key patterns:

• In a linear relationship, equal changes in $x$ (time) give equal changes in $y$ (mass) — the differences in $y$ are constant.
• In a quadratic relationship, the second differences (differences of differences) in $y$ are constant.
• In a relationship where equal changes in $x$ cause $y$ to be multiplied by the same factor (a constant ratio), the values follow a repeated multiplication pattern.

Your table shows:

• Differences in mass are not constant.
• The ratio (factor of $1/2$) is constant.

So the data match a repeated multiplication pattern, not a repeated addition pattern.

A relationship where each step in time multiplies the mass by the same factor (here, $1/2$) is an exponential model. Because the mass is getting smaller over time, it is a decreasing exponential relationship.

From the options given, the function type that best models this data is A) Decreasing exponential.