1. **State the problem:** We are given two points on an exponential decay curve: (0, 30) and (0.5, 26). 2. **Formula for exponential decay:** The general form is $$f(x) = a \cdot b^x$$ where $a$ is the initial value and $0 < b < 1$ for decay. 3. **Identify $a$ and $b$:** Since the point $(0,30)$ is on the curve, $f(0) = a = 30$. 4. **Use the second point to find $b$:** $$f(0.5) = 30 \cdot b^{0.5} = 26$$ Divide both sides by 30: $$b^{0.5} = \frac{26}{30}$$ Square both sides to solve for $b$: $$b = \left(\frac{26}{30}\right)^2$$ 5. **Write the function:** $$f(x) = 30 \cdot \left(\frac{26}{30}\right)^{2x}$$ --- 1. **State the problem:** We are given two points on an exponential growth curve: (0.5, 25) and (1.5, 35). 2. **Formula for exponential growth:** The general form is $$f(x) = a \cdot b^x$$ where $a$ is the initial value and $b > 1$ for growth. 3. **Use the points to find $a$ and $b$:** We do not assume $a$ is the y-intercept. Instead, use the point $(0.5, 25)$: $$f(0.5) = a \cdot b^{0.5} = 25$$ Similarly, for $(1.5, 35)$: $$f(1.5) = a \cdot b^{1.5} = 35$$ 4. **Divide the two equations to eliminate $a$:** $$\frac{f(1.5)}{f(0.5)} = \frac{a b^{1.5}}{a b^{0.5}} = b^{1.5 - 0.5} = b^1 = b = \frac{35}{25} = 1.4$$ 5. **Find $a$ using $b$ and one point:** From $f(0.5) = 25$: $$a \cdot (1.4)^{0.5} = 25$$ Divide both sides: $$a = \frac{25}{(1.4)^{0.5}}$$ 6. **Write the function:** $$f(x) = \frac{25}{(1.4)^{0.5}} \cdot (1.4)^x$$