1. **State the problem:** We are given points from a graph showing bacteria growth over time and need to write an exponential function in the form $f(x) = a b^x$ that fits the data. 2. **Identify points:** From the graph, approximate points are $(1, 4000)$, $(3, 2000)$, $(5, 1000)$, and $(7, 500)$. 3. **Recognize the pattern:** The bacteria count is decreasing, so this is exponential decay, meaning $0 < b < 1$. 4. **Use the general form:** $f(x) = a b^x$ where $a$ is the initial amount and $b$ is the decay factor. 5. **Find $a$:** Using the point $(1, 4000)$, $$4000 = a b^1 = a b$$ 6. **Find $b$:** Using the point $(3, 2000)$, $$2000 = a b^3$$ Divide the two equations to eliminate $a$: $$\frac{2000}{4000} = \frac{a b^3}{a b} = b^{2}$$ $$\Rightarrow \frac{1}{2} = b^{2}$$ $$\Rightarrow b = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \approx 0.707$$ 7. **Find $a$ using $b$ and point $(1,4000)$:** $$4000 = a \times 0.707$$ $$a = \frac{4000}{0.707} \approx 5657.0$$ 8. **Write the function:** $$f(x) = 5657 \times (0.707)^x$$ 9. **Calculate bacteria after 8 hours:** $$f(8) = 5657 \times (0.707)^8$$ Calculate $(0.707)^8$: $$0.707^8 = (0.707^2)^4 = (0.5)^4 = 0.0625$$ So, $$f(8) = 5657 \times 0.0625 = 353.56$$ Rounded to the nearest thousandth: 353.560 **Final answer:** $$f(x) = 5657 \times (0.707)^x$$ Number of bacteria after 8 hours is approximately 353.560.