1. **State the problem:** We have four points $(-2,17)$, $(0,1)$, $(2,1)$, and $(3,22)$ and four function models: cubic $C(x)=ax^3+bx^2+cx+d$, quadratic $Q(x)=ax^2+bx+c$, linear $L(x)=ax+b$, and constant $K(x)=a$. We want to find which model passes through all four points and the predicted value at $x=2.5$ for that model. 2. **Key idea:** A polynomial of degree $n$ can pass through at most $n+1$ points exactly. Since we have 4 points, a cubic polynomial (degree 3) with 4 parameters can fit all points exactly. 3. **Check the models:** - Constant $K(x)=a$ has 1 parameter, cannot fit 4 points. - Linear $L(x)=ax+b$ has 2 parameters, cannot fit 4 points. - Quadratic $Q(x)=ax^2+bx+c$ has 3 parameters, cannot fit 4 points. - Cubic $C(x)=ax^3+bx^2+cx+d$ has 4 parameters, can fit 4 points exactly. 4. **Conclusion:** The cubic model $C(x)$ passes through all four points. 5. **Predicted value at $x=2.5$ for $C(x)$:** Given as 9.125. **Final answer:** The predicted value at $x=2.5$ for the cubic function passing through all points is **9.125**.