1. **State the problem:** We need to graph the quadratic function $$y = -x^2 + 4x - 3$$ and identify its vertex and x-intercepts. 2. **Recall the standard form and formulas:** The quadratic is in the form $$y = ax^2 + bx + c$$ where $$a = -1$$, $$b = 4$$, and $$c = -3$$. - The vertex $$ (h, k) $$ can be found using $$h = -\frac{b}{2a}$$ and $$k = f(h)$$. - The x-intercepts are the solutions to $$0 = ax^2 + bx + c$$, found using the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Find the vertex:** $$h = -\frac{4}{2 \times -1} = -\frac{4}{-2} = 2$$ $$k = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$$ So, the vertex is at $$ (2, 1) $$. 4. **Find the x-intercepts:** Calculate the discriminant: $$\Delta = b^2 - 4ac = 4^2 - 4(-1)(-3) = 16 - 12 = 4$$ Use the quadratic formula: $$x = \frac{-4 \pm \sqrt{4}}{2 \times -1} = \frac{-4 \pm 2}{-2}$$ Calculate each root: For $$+$: $$x = \frac{-4 + 2}{-2} = \frac{-2}{-2} = 1$$ For $$-$$: $$x = \frac{-4 - 2}{-2} = \frac{-6}{-2} = 3$$ So, the x-intercepts are at $$x = 1$$ and $$x = 3$$. 5. **Summary:** - Vertex: $$ (2, 1) $$ - X-intercepts: $$ (1, 0) $$ and $$ (3, 0) $$ This completes the analysis of the quadratic function.