sketch the graph of function "g".  g(x)=2^x +1

The function \( g(x) = 2^x + 1 \) is an exponential function. To sketch its graph, we can follow these steps:

### Key Characteristics of \( g(x) \):
1. **Base Function**: The base function is \( f(x) = 2^x \), which is an exponential function with a base of 2. The graph of \( f(x) = 2^x \) grows rapidly as \( x \) increases.
2. **Vertical Shift**: The "+1" outside of the exponential term shifts the graph of \( 2^x \) vertically upward by 1 unit.

### Steps for Sketching:

1. **Identify the y-intercept**: 
   When \( x = 0 \), 
   \[
   g(0) = 2^0 + 1 = 1 + 1 = 2
   \]
   So, the graph passes through the point \( (0, 2) \).

2. **End behavior**:
   - As \( x \to \infty \), \( 2^x \to \infty \), so \( g(x) \to \infty \).
   - As \( x \to -\infty \), \( 2^x \to 0 \), so \( g(x) \to 1 \). Therefore, the graph approaches the horizontal asymptote \( y = 1 \) from above but never touches it.

3. **Plot a few points**:
   - \( g(1) = 2^1 + 1 = 2 + 1 = 3 \) → Point \( (1, 3) \)
   - \( g(-1) = 2^{-1} + 1 = \frac{1}{2} + 1 = 1.5 \) → Point \( (-1, 1.5) \)
   - \( g(2) = 2^2 + 1 = 4 + 1 = 5 \) → Point \( (2, 5) \)

4. **Asymptote**: The graph has a horizontal asymptote at \( y = 1 \).

### General Behavior:
- The graph starts close to \( y = 1 \) as \( x \to -\infty \).
- It passes through the points \( (0, 2) \), \( (1, 3) \), and grows exponentially as \( x \to \infty \).

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### Relative Questions:
1. What happens to the graph of \( g(x) = 2^x + 1 \) if you change the base to another value, such as 3?
2. How does adding or subtracting a constant inside the exponent \( g(x) = 2^{x+c} \) affect the graph?
3. Can you explain how the graph of \( g(x) = 2^x \) would look without the vertical shift?
4. How would the graph of \( g(x) = -2^x + 1 \) change compared to the current function?
5. What is the effect of multiplying the entire function by a constant, such as \( g(x) = 3(2^x + 1) \)?

**Tip**: When graphing exponential functions, always check for horizontal asymptotes and make sure to calculate key points to guide your sketch.

Sketch the graph of function g. g(x) = 2^x + 1

Learn how to graph the exponential function g(x) = 2^x + 1, including key steps such as identifying the y-intercept, analyzing end behavior, plotting points, and understanding the horizontal asymptote at y = 1. This tutorial is suitable for high school students in Grades 9-12.

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