Compute the differences between consecutive y-values (like $140-200$ and $98-140$). Then also compute the ratios (like $140/200$ and $98/140$). Which one looks more consistent across the table?

Compare $140/200$ and $98/140$. If those two numbers are very close, what does that tell you about how the data are changing from one point to the next?

In Desmos, create a table and enter the x-values $1,2,3,4,5,6$ in the first column and the corresponding y-values $200,140,98,68.6,48,33.6$ in the second column. This lets you see the plotted points.

In a new expression line, type the differences between consecutive y-values, for example: 140-200, 98-140, 68.6-98, 48-68.6, 33.6-48. Look at the outputs and see whether they are all the same number or changing.

In another expression line, type the ratios of consecutive y-values, such as 140/200, 98/140, 68.6/98, 48/68.6, 33.6/48. Examine whether these values are all close to the same number; if they are, that tells you the data follow a repeated-multiplication pattern and helps you choose the correct model type from the options.

Write down the pairs and focus on consecutive y-values:

$(1,200)$, $(2,140)$, $(3,98)$, $(4,68.6)$, $(5,48.0)$, $(6,33.6)$.

As $x$ goes up by $1$ each time, $y$ is going down. Now you need to decide how it is going down: by about the same amount each time, or by about the same factor each time.

Find the differences between consecutive y-values:

• From $200$ to $140$: change is $140-200=-60$
• From $140$ to $98$: change is $98-140=-42$
• From $98$ to $68.6$: change is $68.6-98=-29.4$
• From $68.6$ to $48.0$: change is $48.0-68.6=-20.6$
• From $48.0$ to $33.6$: change is $33.6-48.0=-14.4$

These differences are not equal; they are getting smaller in size. So a model that requires equal changes in $y$ for equal changes in $x$ does not fit well.

Now compare ratios of consecutive y-values:

• $140 \div 200 = 0.7$
• $98 \div 140 = 0.7$
• $68.6 \div 98 \approx 0.7$
• $48.0 \div 68.6 \approx 0.7$
• $33.6 \div 48.0 = 0.7$

Each y-value is about $0.7$ times the previous one. That means as $x$ increases by $1$, $y$ is repeatedly multiplied by roughly the same number (about $0.7$). This is a key sign of a particular type of model.

A model where each step multiplies the previous value by the same constant factor (here, about $0.7$) is called an exponential model, and because the values are going down, it is a decreasing exponential model. Therefore, the best description is Decreasing exponential (choice C).