https://file.aiquickdraw.com/mathbot/file/1726332561154o1obiscv.jpg

**What function is graphed below?**
Possible Answers:
y=|x|/2
y=2(x²)
y=(2^x) + 2

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 \( x \).

This example helps you identify the exponential function y = 2^x + 2 based on its graph. It covers key concepts of exponential growth and graph transformations, making it suitable for students in Grades 9-12.

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