1. **Stating the problem:** We are given data points for voter participation $x$ and government legitimacy $y$ and told the relationship is quadratic of the form $$y = a(x - p)^2 + q$$ with vertex at $(50, 60)$ and another point $(40, 55)$ to find $a$. 2. **Formula and rules:** The vertex form of a quadratic is $$y = a(x - p)^2 + q$$ where $(p, q)$ is the vertex. Given vertex $(p, q) = (50, 60)$, the model is: $$y = a(x - 50)^2 + 60$$ 3. **Use the point $(40, 55)$ to find $a$:** Substitute $x=40$, $y=55$: $$55 = a(40 - 50)^2 + 60$$ $$55 = a(-10)^2 + 60$$ $$55 = 100a + 60$$ 4. **Solve for $a$:** $$55 - 60 = 100a$$ $$-5 = 100a$$ $$a = \frac{-5}{100}$$ Intermediate step with cancellation: $$a = \frac{\cancel{-5}}{\cancel{100}} = -\frac{1}{20}$$ 5. **Final quadratic model:** $$y = -\frac{1}{20}(x - 50)^2 + 60$$ This model describes the quadratic relationship between voter participation and government legitimacy with vertex at $(50,60)$ and passing through $(40,55)$.