Quadratic Graph E9E7D0

1. **State the problem:** We need to identify the graph of the quadratic function $$f(x) = 3x^2 + 18x + 25$$. 2. **Formula and rules:** The graph of a quadratic function $$f(x) = ax^2 + bx + c$$ is a parabola. The vertex form is useful to find the vertex: $$f(x) = a(x - h)^2 + k$$ where $$(h, k)$$ is the vertex. 3. **Find the vertex:** The vertex $$h$$ is given by $$h = -\frac{b}{2a}$$. Here, $$a = 3$$ and $$b = 18$$. Calculate: $$h = -\frac{18}{2 \times 3} = -\frac{18}{6} = -3$$. 4. **Find $$k$$ by substituting $$h$$ back into $$f(x)$$:** $$k = f(-3) = 3(-3)^2 + 18(-3) + 25 = 3(9) - 54 + 25 = 27 - 54 + 25 = -2$$. 5. **Vertex and axis of symmetry:** - Vertex: $$(-3, -2)$$ - Axis of symmetry: $$x = -3$$ 6. **Direction of parabola:** Since $$a = 3 > 0$$, the parabola opens upward. 7. **Conclusion:** The parabola has vertex $$(-3, -2)$$, axis of symmetry $$x = -3$$, and opens upward. This matches the description of the **top-right** graph. **Final answer:** The graph is the upward-opening parabola with vertex at $$(-3, -2)$$, axis of symmetry $$x = -3$$, crossing near $$y = -2$$.