1. **State the problem:** We need to find a function rule that relates the number of minutes $m$ to the number of bacteria $b(m)$ in the population. 2. **Analyze the data points:** The data points given are approximately $(0,0)$, $(1,2)$, $(2,8)$, $(3,16)$, $(4,60)$, and $(5,160)$. 3. **Identify the type of function:** The problem states the graph is an increasing exponential curve, so we assume a function of the form: $$b(m) = a \cdot r^m$$ where $a$ is the initial amount and $r$ is the growth rate. 4. **Use the initial point:** At $m=0$, $b(0) = a \cdot r^0 = a = 0$ according to the point $(0,0)$, but an exponential function cannot have zero initial value. This suggests the data might be shifted or the zero point is approximate. 5. **Use the next points to find $a$ and $r$:** Using $(1,2)$ and $(2,8)$: $$b(1) = a r = 2$$ $$b(2) = a r^2 = 8$$ Divide the second equation by the first: $$\frac{a r^2}{a r} = \frac{8}{2} \Rightarrow r = 4$$ Then from $b(1) = a r = 2$: $$a \cdot 4 = 2 \Rightarrow a = \frac{2}{4} = 0.5$$ 6. **Write the function:** $$b(m) = 0.5 \cdot 4^m$$ 7. **Check with other points:** At $m=3$: $$b(3) = 0.5 \cdot 4^3 = 0.5 \cdot 64 = 32$$ The data point is 16, so the model is approximate but captures the exponential growth trend. **Final answer:** $$b(m) = 0.5 \cdot 4^m$$