To graph the function \( f(x) = 2^x + 1 \), follow these steps:

1. **Identify the type of function**: This is an exponential function where the base is 2, and the graph is shifted vertically by +1.

2. **Key points to plot**:
   - When \( x = 0 \), \( f(x) = 2^0 + 1 = 1 + 1 = 2 \).
   - When \( x = 1 \), \( f(x) = 2^1 + 1 = 2 + 1 = 3 \).
   - When \( x = -1 \), \( f(x) = 2^{-1} + 1 = \frac{1}{2} + 1 = 1.5 \).
   - When \( x = 2 \), \( f(x) = 2^2 + 1 = 4 + 1 = 5 \).
   - As \( x \to -\infty \), \( f(x) \to 1 \), which represents the horizontal asymptote at \( y = 1 \).

3. **General shape**:
   - The graph is increasing for all \( x \), as it's an exponential growth function.
   - The horizontal asymptote is at \( y = 1 \), meaning the function approaches 1 but never touches it as \( x \to -\infty \).
   - The function crosses the point \( (0, 2) \) and grows rapidly as \( x \to \infty \).

4. **Domain and range**:
   - Domain: \( (-\infty, \infty) \) (all real numbers)
   - Range: \( (1, \infty) \) (the function value is always greater than 1)

### Would you like to see a plotted graph or have any questions?

Here are 5 related questions to deepen your understanding:
1. What happens to the graph of \( f(x) = 2^x + 1 \) if we change the base from 2 to 3?
2. How does the vertical shift affect the graph of an exponential function?
3. What is the effect of a negative exponent on the value of an exponential function?
4. How would the graph change if we subtracted 1 instead of adding 1 in the function \( f(x) = 2^x + 1 \)?
5. What are the characteristics of the horizontal asymptote for other exponential functions?

**Tip**: To better understand exponential functions, try plotting different values for the base and observe how they influence the growth rate and overall shape.

Learn how to graph the exponential function f(x) = 2^x + 1 with key points, a horizontal asymptote, and a detailed explanation of exponential growth and vertical shifts. This concept is ideal for high school students in Grades 9-12.

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