1. **Problem statement:** Determine if the function $f(x) = a \cdot q^x$ with points $A(3|2)$ and $B(8|5)$ represents exponential growth. 2. **Recall the formula:** The general form of an exponential function is $$f(x) = a \cdot q^x$$ where $a$ is the initial value and $q$ is the growth factor. 3. **Given points:** $f(3) = 2$ and $f(8) = 5$. 4. **Set up equations:** $$2 = a \cdot q^3$$ $$5 = a \cdot q^8$$ 5. **Divide the second equation by the first to eliminate $a$:** $$\frac{5}{2} = \frac{a \cdot q^8}{a \cdot q^3} = q^{8-3} = q^5$$ 6. **Solve for $q$:** $$q^5 = \frac{5}{2}$$ $$q = \sqrt[5]{\frac{5}{2}} \approx 1.197$$ 7. **Interpretation:** Since $q \approx 1.197 > 1$, the function represents exponential growth. 8. **Find $a$ using $f(3) = 2$:** $$2 = a \cdot (1.197)^3$$ $$a = \frac{2}{(1.197)^3} \approx \frac{2}{1.716} \approx 1.166$$ 9. **Final function:** $$f(x) = 1.166 \cdot (1.197)^x$$ **Answer:** The function is an exponential growth function with growth factor approximately $1.197$.