Parabola Maximum Zeros A65904

1. **State the problem:** We are given the quadratic function $$f(x) = -\frac{1}{2} x^2 - x - \frac{7}{2}$$ and its graph, a downward-opening parabola with vertex at $(2, -2)$. 2. **Find the maximum value of the function:** Since the parabola opens downward (coefficient of $x^2$ is negative), the vertex represents the maximum point. 3. **Maximum value from the equation:** The vertex form of a parabola is $$f(x) = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. Given the vertex at $(2, -2)$, the maximum value is $$k = -2$$. 4. **Maximum value from the graph:** The graph shows the vertex at $(2, -2)$, so the maximum value from the graph is also $$-2$$. 5. **Conclusion on maximum values:** Both the equation and the graph have the same maximum value of $$-2$$. 6. **Check for zeros (roots) of the function:** To find zeros, solve $$f(x) = 0$$: $$-\frac{1}{2} x^2 - x - \frac{7}{2} = 0$$ Multiply both sides by $-2$ to clear fractions: $$x^2 + 2x + 7 = 0$$ 7. **Calculate discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times 7 = 4 - 28 = -24$$ Since $$\Delta < 0$$, there are no real roots. 8. **Conclusion on zeros:** Both the equation and the graph have no zeros. **Final answers:** - The equation and the graph have the same maximum value of $$-2$$. - Both the equation and the graph have no zeros.