1. **Problem statement:** Given tables of values for two exponential functions, find the constant ratio, initial value, and write the exponential function $f(x)$ for each. --- ### a. 2. **Identify initial value:** The initial value is $f(0)$, which is 4. 3. **Find constant ratio:** Calculate the ratio between consecutive values: $$\frac{f(1)}{f(0)} = \frac{2}{4} = \frac{1}{2}$$ Check next ratio: $$\frac{f(2)}{f(1)} = \frac{1}{2} = \frac{1}{2}$$ So the constant ratio $r = \frac{1}{2}$. 4. **Write the exponential function:** $$f(x) = a \cdot r^x$$ where $a$ is the initial value and $r$ is the constant ratio. So, $$f(x) = 4 \cdot \left(\frac{1}{2}\right)^x$$ --- ### b. 5. **Identify initial value:** $f(0) = 3$ 6. **Find constant ratio:** $$\frac{f(1)}{f(0)} = \frac{6}{3} = 2$$ Check next ratio: $$\frac{f(2)}{f(1)} = \frac{12}{6} = 2$$ So the constant ratio $r = 2$. 7. **Write the exponential function:** $$f(x) = 3 \cdot 2^x$$ --- **Final answers:** - a. $f(x) = 4 \cdot \left(\frac{1}{2}\right)^x$ - b. $f(x) = 3 \cdot 2^x$