It looks like you are working with a quadratic equation in vertex form:

$y = \frac{1}{2} \left(x + 4\right)^2 + 3$

To identify whether this quadratic function has a maximum or minimum value and what that value is, we can analyze the given function.

Key Observations:

1. Vertex Form of Quadratic Function: The given equation is in the vertex form: $y = a(x - h)^2 + k$ where the vertex of the parabola is at $(h, k)$.

2. Coefficient $a$: The coefficient $a = \frac{1}{2}$ is positive, meaning the parabola opens upwards, and hence, the function has a minimum value.

3. Vertex: In this equation, $h = -4$ and $k = 3$, so the vertex is at $(-4, 3)$.

4. Minimum Value: Since the parabola opens upwards, the minimum value of $y$ occurs at the vertex, which is $y = 3$.

Conclusion:

• The quadratic function has a minimum value.
• The minimum value of the function is 3.

Would you like to explore further details about this or need assistance with other concepts?

Here are 5 related questions you might find interesting:

1. How do you convert a quadratic function from standard form to vertex form?
2. What happens to the graph of a quadratic function if the coefficient $a$ is negative?
3. How do you find the x-intercepts (roots) of a quadratic function?
4. What is the significance of the axis of symmetry in a quadratic function?
5. How can you graph a quadratic function using its vertex and additional points?

Tip: When working with quadratic functions, always look at the sign of $a$ to determine if the parabola opens upwards (minimum) or downwards (maximum).