One function type fits data where the y-value grows by the same amount each step, another where the second differences are about the same, and another where the y-value grows by about the same percentage or factor each step. Which pattern seems closest here?

Try dividing each colony size by the previous one (for example, $3.1/2.0$, then $4.8/3.1$). Are those ratios roughly the same number? That points you toward one particular type of model.

Create a table in Desmos. In the first column (say $x_1$), enter the times: 0, 1, 2, 3, 4. In the second column ($y_1$), enter the colony sizes: 2, 3.1, 4.8, 7.3, 11.1. This will display the scatterplot of the data.

In a new expression line, type y1 ~ m x1 + b. Desmos will find a best-fit line. Look at how closely this line matches the plotted points—especially at the higher times.

In another expression line, type y1 ~ a x1^2 + b x1 + c. Compare this curve to the scatterplot. See whether it goes through or near all the points or if it misses especially at the ends.

In a new line, type y1 ~ a b^x1. Desmos will fit an exponential curve to the data. Compare this curve’s shape and closeness to the points with the linear and quadratic fits from the previous steps.

You can type an equation like y = k/x using a slider for $k$ to see the general shape of an inverse variation. Compare that shape to your scatterplot—pay attention to whether it increases or decreases as $x$ increases and whether it can even pass through a point with $x = 0$.

For a linear model, the difference between consecutive y-values (colony sizes) should be roughly constant.

Compute the differences:

• From 0 to 1 hour: $3.1 - 2.0 = 1.1$
• From 1 to 2 hours: $4.8 - 3.1 = 1.7$
• From 2 to 3 hours: $7.3 - 4.8 = 2.5$
• From 3 to 4 hours: $11.1 - 7.3 = 3.8$

These increases are not close to constant; they are getting larger each time, so a linear function is not a good fit.

For a quadratic model, the second differences (the differences of the differences) should be roughly constant.

We already have the first differences: $1.1$, $1.7$, $2.5$, $3.8$. Now find the second differences:

• Between $1.1$ and $1.7$: $1.7 - 1.1 = 0.6$
• Between $1.7$ and $2.5$: $2.5 - 1.7 = 0.8$
• Between $2.5$ and $3.8$: $3.8 - 2.5 = 1.3$

These second differences are not close to constant, so a quadratic function is unlikely to be the best model.

For an inverse variation model, the product $x \cdot y$ is roughly constant and $y$ usually decreases as $x$ increases.

Here, as time increases from 0 to 4 hours, the colony size increases instead of decreasing. That behavior does not match inverse variation, so we can rule out that option as well.

When data are best modeled by an exponential function, the y-values change by about the same factor (ratio) each step, not the same difference.

Compute the ratios of consecutive colony sizes:

• From 0 to 1 hour: $\dfrac{3.1}{2.0} \approx 1.55$
• From 1 to 2 hours: $\dfrac{4.8}{3.1} \approx 1.55$
• From 2 to 3 hours: $\dfrac{7.3}{4.8} \approx 1.52$
• From 3 to 4 hours: $\dfrac{11.1}{7.3} \approx 1.52$

These ratios are all around $1.5$, meaning the colony is growing by roughly the same factor each hour. That pattern is best modeled by an exponential function.