From the graph you've provided, the function exhibits exponential growth, particularly for values of $x > 0$. Let's analyze the possible answers:

1. $y = \frac{|x|}{2}$: This represents a V-shaped graph (absolute value function), which doesn't match the graph shown. The graph in your image does not show a sharp turn like an absolute value function.

2. $y = 2x^2$: This represents a parabola, which would be symmetric about the y-axis. The graph you're showing is not symmetric and instead shows a rapid increase for positive values of $x$, suggesting it's not a quadratic function.

3. $y = 2^x + 2$: This is an exponential function shifted vertically by 2 units. It starts close to $y = 2$ for negative $x$ values and then grows rapidly as $x$ increases, which matches the behavior seen in the graph.

Based on the appearance of the curve, the correct function is:

$y = 2^x + 2$

Would you like further details or have any questions?

Here are five related questions:

1. What are the properties of exponential functions in general?
2. How does shifting a function vertically affect its graph?
3. Can you derive the equation of the function from points on the graph?
4. What is the domain and range of the function $y = 2^x + 2$?
5. How does the growth rate of an exponential function compare to a polynomial function like $y = 2x^2$?

Tip: Exponential functions grow much faster than polynomial functions for large values of $x$.