1. **State the problem:** Simplify and rewrite the function $y = -3\left(\frac{1}{2}x^2 - 2\right) + \frac{15}{2}$. 2. **Apply the distributive property:** Multiply $-3$ by each term inside the parentheses. $$y = -3 \times \frac{1}{2}x^2 + (-3) \times (-2) + \frac{15}{2}$$ 3. **Calculate each multiplication:** $$y = -\frac{3}{2}x^2 + 6 + \frac{15}{2}$$ 4. **Combine the constant terms:** Convert $6$ to a fraction with denominator $2$ to add easily. $$6 = \frac{12}{2}$$ So, $$y = -\frac{3}{2}x^2 + \frac{12}{2} + \frac{15}{2}$$ 5. **Add the fractions:** $$\frac{12}{2} + \frac{15}{2} = \frac{27}{2}$$ 6. **Final simplified function:** $$y = -\frac{3}{2}x^2 + \frac{27}{2}$$ This is a quadratic function opening downward with vertex at the maximum point.