1. **Problem:** Graph and analyze the quadratic function $$f(x) = -x^2 - 2x + 3$$. 2. **Formula and rules:** The vertex form of a quadratic is $$f(x) = a(x-h)^2 + k$$ where $$(h,k)$$ is the vertex. 3. **Find the vertex:** Use $$h = -\frac{b}{2a}$$ for $$f(x) = ax^2 + bx + c$$. Here, $$a = -1$$, $$b = -2$$, $$c = 3$$. Calculate $$h = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1$$. 4. **Find $$k$$:** $$k = f(-1) = -(-1)^2 - 2(-1) + 3 = -1 + 2 + 3 = 4$$. 5. **Vertex:** $$(-1, 4)$$. 6. **Rewrite in vertex form:** $$f(x) = - (x + 1)^2 + 4$$. 7. **Find y-intercept:** Set $$x=0$$: $$f(0) = -0 - 0 + 3 = 3$$. So y-intercept is $$(0, 3)$$. 8. **Axis of symmetry:** $$x = -1$$. 9. **Graph shape:** Since $$a = -1 < 0$$, the parabola opens downward. 10. **Summary:** Vertex at $$(-1,4)$$, opens downward, y-intercept at $$(0,3)$$, axis of symmetry $$x = -1$$. **Final answer:** The quadratic function $$f(x) = -x^2 - 2x + 3$$ has vertex $$(-1,4)$$, opens downward, and y-intercept at $$(0,3)$$.