To find the minimum value of the function $y = x^2 + 3$, let's analyze it step by step.

1. The equation $y = x^2 + 3$ represents a parabola. In general, a quadratic function in the form $y = ax^2 + bx + c$ opens upwards if $a > 0$, which is the case here since $a = 1$.

2. When a parabola opens upwards, its minimum value occurs at the vertex. For the function $y = x^2 + 3$, the vertex is at $x = 0$ because there is no $bx$ term (i.e., $b = 0$).

3. Substitute $x = 0$ into the equation to find the minimum $y$-value: $y = (0)^2 + 3 = 3$

Thus, the minimum value of $y = x^2 + 3$ is 3.

Would you like a more detailed explanation of any steps or have further questions?

Here are some related questions that build on this topic:

1. How do you find the vertex of a parabola for functions with a non-zero $b$-term?
2. Why does the parabola open upwards in this function?
3. What would the minimum value be if the equation was $y = x^2 - 5$?
4. How do you determine the axis of symmetry for a quadratic function?
5. Can a quadratic function have a maximum value, and if so, under what conditions?

Tip: For quadratic functions in the form $y = ax^2 + bx + c$, the vertex formula $x = -\frac{b}{2a}$ is essential to determine the maximum or minimum values.