what is the horizontal asymptote of y=2^x+b

The horizontal asymptote of the function \( y = 2^x + b \) is determined by analyzing the behavior of the function as \( x \) approaches infinity or negative infinity.

- As \( x \) approaches infinity (\( x \to \infty \)), \( 2^x \) grows exponentially large, so \( y = 2^x + b \) also grows without bound, meaning there is no horizontal asymptote in this direction.

- As \( x \) approaches negative infinity (\( x \to -\infty \)), \( 2^x \) approaches 0. Therefore, \( y = 2^x + b \) approaches \( y = 0 + b = b \).

Thus, the horizontal asymptote of the function \( y = 2^x + b \) is \( y = b \).

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Here are 5 related questions:
1. How do you find vertical asymptotes in other types of functions?
2. What is the general form of a function with an exponential component?
3. How does changing the base of the exponential function affect its graph?
4. What happens to the horizontal asymptote if a coefficient is added to \( 2^x \) instead of a constant?
5. Can a function have more than one horizontal asymptote?

**Tip:** Horizontal asymptotes describe the end behavior of a function, indicating what value \( y \) approaches as \( x \) goes to infinity or negative infinity.

Explore the concept of horizontal asymptotes in exponential functions with the equation y = 2^x + b. Learn how the function behaves as x approaches infinity and negative infinity, determining the horizontal asymptote value as y = b. Suitable for high school students studying exponential functions and asymptotes.

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