1. **State the problem:** We are given the function $h(x) = (x + 3)^2 + 3$ and need to analyze its graph. 2. **Formula and rules:** This is a quadratic function in vertex form $h(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex and $a$ determines the direction of the parabola. 3. **Identify vertex and direction:** Here, $a = 1 > 0$, so the parabola opens upwards. 4. The vertex is at $(-3, 3)$ because the function is $(x - (-3))^2 + 3$. 5. **Axis of symmetry:** The vertical line through the vertex is $x = -3$. 6. **Summary:** The graph is a parabola opening upwards with vertex at $(-3, 3)$ and axis of symmetry $x = -3$. Final answer: Vertex $(-3, 3)$, parabola opens upwards, axis of symmetry $x = -3$.