1. **Stating the problem:** We want to find a quadratic function $f(x) = ax^2 + bx + c$ that fits the given temperature data over time without using the second difference formula. 2. **Use three points to create equations:** Since a quadratic has three unknowns ($a$, $b$, $c$), we can use three data points to form three equations. Choose points at $x=0$, $x=1$, and $x=2$: \begin{align*} f(0) &= c = 10 \\ f(1) &= a(1)^2 + b(1) + c = a + b + c = 14.5 \\ f(2) &= a(2)^2 + b(2) + c = 4a + 2b + c = 20 \end{align*} 3. **Substitute $c=10$ into the other equations:** \begin{align*} a + b + 10 &= 14.5 \\ 4a + 2b + 10 &= 20 \end{align*} Simplify: \begin{align*} a + b &= 4.5 \\ 4a + 2b &= 10 \end{align*} 4. **Solve the system of equations:** Multiply the first equation by 2: $$2a + 2b = 9$$ Subtract this from the second equation: $$4a + 2b - (2a + 2b) = 10 - 9$$ $$2a = 1 \\ a = 0.5$$ 5. **Find $b$:** Substitute $a=0.5$ into $a + b = 4.5$: $$0.5 + b = 4.5 \\ b = 4$$ 6. **Write the final quadratic function:** $$f(x) = 0.5x^2 + 4x + 10$$ **Answer:** $f(x) = 0.5x^2 + 4x + 10$