1. **Stating the problem:** We want to understand the graph of a quadratic function of the form $$y = x^2 + bx + c$$ where $$b$$ and $$c$$ are integers (elements of $$\mathbb{Z}$$). 2. **Formula and important rules:** The quadratic function is given by $$y = ax^2 + bx + c$$ with $$a=1$$ here. The graph is a parabola opening upwards because $$a > 0$$. 3. **Vertex form:** To analyze the graph, convert to vertex form using completing the square: $$y = x^2 + bx + c = (x^2 + bx) + c$$ $$= \left(x^2 + bx + \left(\frac{b}{2}\right)^2\right) - \left(\frac{b}{2}\right)^2 + c$$ $$= \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c$$ 4. **Vertex coordinates:** The vertex is at $$\left(-\frac{b}{2}, c - \frac{b^2}{4}\right)$$. 5. **Axis of symmetry:** The vertical line $$x = -\frac{b}{2}$$ is the axis of symmetry. 6. **Y-intercept:** When $$x=0$$, $$y = c$$, so the graph crosses the y-axis at $$(0,c)$$. 7. **X-intercepts:** Solve $$x^2 + bx + c = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$$ The discriminant $$\Delta = b^2 - 4c$$ determines the number of real roots: - If $$\Delta > 0$$, two distinct real roots. - If $$\Delta = 0$$, one real root (vertex on x-axis). - If $$\Delta < 0$$, no real roots. 8. **Summary:** The graph is a parabola opening upwards with vertex at $$\left(-\frac{b}{2}, c - \frac{b^2}{4}\right)$$, y-intercept at $$(0,c)$$, and x-intercepts depending on the discriminant. **Final answer:** The quadratic function $$y = x^2 + bx + c$$ with integer $$b,c$$ graphs as a parabola with vertex $$\left(-\frac{b}{2}, c - \frac{b^2}{4}\right)$$ and axis of symmetry $$x = -\frac{b}{2}$$.