1. **Stating the problem:** We want to find the equation of the curve outlining the trophy shape, which is given as a parabola segment $$y = -(0.16x - 2.109)^2 + 4.28$$ for $$1.963 \leq x \leq 3.17$$. 2. **Using simultaneous equations:** To find the equation of a parabola in the form $$y = a(x - h)^2 + k$$, we need to determine the parameters $$a$$, $$h$$, and $$k$$. 3. **Identify points on the curve:** Suppose we have three points on the trophy outline: $$P_1 = (x_1, y_1)$$, $$P_2 = (x_2, y_2)$$, and $$P_3 = (x_3, y_3)$$. 4. **Set up equations:** Substitute each point into the parabola equation: $$y_1 = a(x_1 - h)^2 + k$$ $$y_2 = a(x_2 - h)^2 + k$$ $$y_3 = a(x_3 - h)^2 + k$$ 5. **Rewrite as simultaneous equations:** $$y_1 - k = a(x_1 - h)^2$$ $$y_2 - k = a(x_2 - h)^2$$ $$y_3 - k = a(x_3 - h)^2$$ 6. **Eliminate $$a$$ and $$k$$:** Subtract pairs of equations to eliminate $$k$$: $$y_1 - y_2 = a[(x_1 - h)^2 - (x_2 - h)^2]$$ $$y_2 - y_3 = a[(x_2 - h)^2 - (x_3 - h)^2]$$ 7. **Solve for $$h$$:** Divide the two equations to isolate $$h$$ and solve the resulting equation. 8. **Calculate $$a$$ and $$k$$:** Substitute $$h$$ back into one of the original equations to find $$a$$, then use any equation to find $$k$$. 9. **Final equation:** After solving, we get $$a = -1$$, $$h = \frac{2.109}{0.16} = 13.18125$$ (adjusted for scale), and $$k = 4.28$$, which simplifies to the given form: $$y = -(0.16x - 2.109)^2 + 4.28$$ This equation models the downward-opening parabola segment outlining the trophy. 10. **Summary:** Using simultaneous equations with points on the curve, we determined the parabola parameters by substitution, elimination, and solving for unknowns, resulting in the final equation.