Question: Consider the quadratic function $y = -x^2 + 6x - 5$. What do we know about the graph of this quadratic equation, based on its formula? The vertex is (3,-32) and it opens upward. The vertex is (3,4) and it opens downward. The vertex is (3,-32) and it opens downward. The vertex is (3,4) and it opens upward.
1. **State the problem:** We are given the quadratic function $$y = -x^2 + 6x - 5$$ and need to determine the vertex and the direction the parabola opens.
2. **Recall the vertex formula:** For a quadratic function $$y = ax^2 + bx + c$$, the vertex $$x$$-coordinate is given by $$x = -\frac{b}{2a}$$.
3. **Identify coefficients:** Here, $$a = -1$$, $$b = 6$$, and $$c = -5$$.
4. **Calculate the vertex $$x$$-coordinate:**
$$x = -\frac{6}{2 \times -1} = -\frac{6}{-2} = 3$$
5. **Calculate the vertex $$y$$-coordinate:** Substitute $$x=3$$ into the function:
$$y = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4$$
6. **Determine the direction the parabola opens:** Since $$a = -1 < 0$$, the parabola opens downward.
7. **Conclusion:** The vertex is $$(3,4)$$ and the parabola opens downward.
**Final answer:** The vertex is $$(3,4)$$ and it opens downward.