If the function were linear, the amount would go down by the same number of milligrams each hour. Does that match what the table shows?

Try dividing each amount by the previous one (for example, $60/80$, $45/60$, and so on). Are those quotients more similar to each other than the differences are?

All of the function types listed can model curves, but some increase over time and some decrease. Think about whether the data in the table are going up or going down.

Create a table in Desmos. In the first column (say $x_1$), enter the times $0, 1, 2, 3, 4, 5$. In the second column (say $y_1$), enter the corresponding amounts $80, 60, 45, 34, 26, 19$.

Look at the scatterplot formed by the points $(x_1, y_1)$. Decide whether they lie close to a straight line, form a parabolic (U-shaped) curve, or bend smoothly downward in a way that suggests a multiplicative pattern.

In the expression lines, you can try different regressions: type y1 ~ m x1 + b for linear, y1 ~ a x1^2 + b x1 + c for quadratic, and y1 ~ a b^{x1} for exponential. Compare which curve matches the plotted points most closely; then match that model type (linear, quadratic, or exponential, and whether it represents increase or decrease) to the best answer choice.

List the amounts and how much they change from one hour to the next:

• Amounts: 80, 60, 45, 34, 26, 19
• First differences (change each hour):
  • $60 - 80 = -20$
  • $45 - 60 = -15$
  • $34 - 45 = -11$
  • $26 - 34 = -8$
  • $19 - 26 = -7$

These changes are not equal, so the amount does not decrease by a constant amount each hour.

A linear function has a constant rate of change, meaning the first differences between $y$-values (amounts) are the same each step.

Here, the first differences are $-20, -15, -11, -8, -7$, which are not equal.

So the situation cannot be modeled well by a linear function with a constant rate of decrease (choice A).

For equally spaced $x$-values (time), a quadratic function has second differences that are constant.

From the first differences $-20, -15, -11, -8, -7$, find the second differences:

• $-15 - (-20) = 5$
• $-11 - (-15) = 4$
• $-8 - (-11) = 3$
• $-7 - (-8) = 1$

The second differences are $5, 4, 3, 1$, which are not equal, so a quadratic model (choice B) does not fit well either.

When $x$ increases by the same amount each time (here, 1 hour), an exponential relationship shows up as almost constant ratios between consecutive $y$-values.

Compute the ratios of each amount to the previous one:

• $60/80 = 0.75$
• $45/60 = 0.75$
• $34/45 \approx 0.76$
• $26/34 \approx 0.76$
• $19/26 \approx 0.73$

These ratios are all close to about $0.75$. That means each hour you multiply by roughly the same factor, so the amount is changing by about the same percentage each hour, not by the same absolute amount.

Because the amounts are decreasing over time and the ratios between consecutive amounts are roughly constant, the data are best modeled by an exponential decay function with a constant percent decrease, which corresponds to answer choice C.