For exponential growth, a common form is $m = a\cdot b^t$. Use the value from the table at $t=0$ to figure out $a$.

To estimate the growth factor $b$ in an exponential model, compare $\dfrac{m_{t+1}}{m_t}$ for several steps. What do these ratios suggest $b$ should be close to?

Once you know the starting value and an approximate growth factor, see which option uses that starting value and a base (the number raised to $t$) closest to your estimate.

In Desmos, create a table (using the "+" button and choosing "Table") and enter the time values $0,1,2,3,4$ in the first column and the corresponding masses $2.0, 2.9, 4.3, 6.0, 8.8$ in the second column. This will plot the data points on the graph.

In separate lines, type each equation using $x$ for time and $y$ for mass:

• $y = 2(1.8)^x$
• $y = 2 + 1.9x$
• $y = 2(1.5)^x$
• $y = 1.2(2)^x$ Make sure all four graphs are turned on (checkboxes checked).

Look at which graph passes through or closest to the plotted data points at $x=0,1,2,3,4$. Focus especially on matching both the starting value at $x=0$ and the overall curved shape of the data. The equation whose curve best follows the points is the model you should choose.

Compute the differences between consecutive masses:

• From $t=0$ to $t=1$: $2.9 - 2.0 = 0.9$
• From $t=1$ to $t=2$: $4.3 - 2.9 = 1.4$
• From $t=2$ to $t=3$: $6.0 - 4.3 = 1.7$
• From $t=3$ to $t=4$: $8.8 - 6.0 = 2.8$

These increases are not constant, so the data are not well modeled by a linear function like $m = 2 + 1.9t$. The problem also says the scatterplot suggests exponential growth, so we should use an exponential model, not a linear one.

A simple exponential growth model has the form

where $a$ is the initial value (mass at $t=0$) and $b$ is the growth factor per hour.

From the table, when $t=0$, $m = 2.0$. Plugging $t=0$ into $m = a\cdot b^t$ gives

So $a = 2$. This means any good exponential model should start with $2$ grams at $t=0$, which rules out any model whose value at $t=0$ is not $2$.

For exponential growth, the ratios $\dfrac{m_{t+1}}{m_t}$ should be roughly constant.

Compute these ratios:

• From $t=0$ to $t=1$: $\dfrac{2.9}{2.0} \approx 1.45$
• From $t=1$ to $t=2$: $\dfrac{4.3}{2.9} \approx 1.48$
• From $t=2$ to $t=3$: $\dfrac{6.0}{4.3} \approx 1.40$
• From $t=3$ to $t=4$: $\dfrac{8.8}{6.0} \approx 1.47$

These are all around $1.4$ to $1.5$, so the culture multiplies by about $1.5$ each hour, not by $1.8$ or $2$. The best exponential model will therefore have a base (growth factor) close to $1.5$.

We found that a reasonable exponential model should:

• Start at $2$ grams when $t=0$, and
• Multiply by about $1.5$ each hour.

The equation that does this is

Check it against the data:

• At $t=1$: $m = 2(1.5)^1 = 3.0$ (close to $2.9$)
• At $t=2$: $m = 2(1.5)^2 = 2(2.25) = 4.5$ (close to $4.3$)
• At $t=3$: $m = 2(1.5)^3 = 2(3.375) = 6.75$ (close to $6.0$)
• At $t=4$: $m = 2(1.5)^4 = 2(5.0625) \approx 10.1$ (a bit above $8.8$, but still fits the exponential trend well)

Among the choices given, this equation best matches an exponential pattern with the correct initial value and approximate growth factor, so $m = 2(1.5)^t$ is the correct model.