https://file.aiquickdraw.com/mathbot/file/172913872628072zprt8e.jpg

The function given in the image is:

\[
g(x) = 2 + e^x
\]

This function is a combination of a constant (2) and the exponential function \( e^x \). Here's a step-by-step explanation of how to graph this function:

### 1. General Shape of \( e^x \):
- The graph of \( e^x \) is an exponential growth function.
- It is always positive, and as \( x \) increases, \( e^x \) grows rapidly.
- As \( x \) becomes more negative, \( e^x \) approaches 0 but never touches the x-axis.

### 2. Shifting by 2:
- The function \( g(x) = 2 + e^x \) shifts the graph of \( e^x \) upwards by 2 units.
- The y-intercept (where \( x = 0 \)) will be at \( g(0) = 2 + e^0 = 2 + 1 = 3 \).

### 3. Key Points:
- When \( x = 0 \), \( g(x) = 3 \).
- As \( x \to -\infty \), \( e^x \to 0 \), so \( g(x) \) approaches 2.
- As \( x \to \infty \), \( e^x \to \infty \), so \( g(x) \) increases without bound.

### 4. Horizontal Asymptote:
- The horizontal asymptote is at \( y = 2 \) because \( g(x) = 2 + e^x \), and as \( e^x \to 0 \) for large negative \( x \), the value of \( g(x) \) approaches 2.

### 5. Graph Characteristics:
- The curve is always above the line \( y = 2 \).
- The slope of the curve becomes steeper as \( x \) increases, reflecting the exponential growth.

Would you like me to explain any of these points in more detail or assist in graphing this function step by step?

Here are some related questions for further exploration:
1. How do you find the derivative of \( g(x) = 2 + e^x \) and what does it say about the slope?
2. What happens if you add a negative constant to the exponential function?
3. How would the graph change if the function were \( g(x) = 2 - e^x \)?
4. What is the inverse of the function \( g(x) = 2 + e^x \)?
5. How do you find the tangent line to the graph at \( x = 0 \)?

**Tip**: When graphing exponential functions, always note the horizontal asymptote and how the graph behaves for large positive and negative values of \( x \).

Learn how to graph the exponential function g(x) = 2 + e^x, covering the key points of exponential growth, graph shifting, and horizontal asymptotes. Understand its behavior for large positive and negative x values. Suitable for students in Grades 10-12.

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