Create two different exponential functions of the form f (X)=a*b^x+c that have a horizontal asymptote at y=5

To create two different exponential functions of the form \( f(x) = a \cdot b^x + c \) that have a horizontal asymptote at \( y = 5 \), we need to ensure that the constant \( c \) in the equation is 5. This is because, as \( x \) approaches infinity, the term \( a \cdot b^x \) will approach 0 if \( 0 < b < 1 \), and the function will asymptotically approach \( c \).

Here are two different exponential functions with this property:

### Function 1:
\[ f(x) = 3 \cdot (0.5)^x + 5 \]

### Function 2:
\[ f(x) = -2 \cdot (0.8)^x + 5 \]

#### Explanation:
- **Function 1:** Here, \( a = 3 \), \( b = 0.5 \), and \( c = 5 \). As \( x \) increases, \( (0.5)^x \) becomes smaller and smaller, making \( f(x) \) approach 5.
- **Function 2:** Here, \( a = -2 \), \( b = 0.8 \), and \( c = 5 \). Similarly, as \( x \) increases, \( (0.8)^x \) becomes smaller, and the function approaches the asymptote at 5.

Would you like more details or have any questions?

Here are 5 related questions to expand on this information:
1. How do different values of \( a \) affect the shape of the exponential function?
2. What happens to the function if \( b > 1 \)?
3. How can we determine the y-intercept of these functions?
4. What is the significance of the base \( b \) in the exponential function?
5. How can you find the exact point where the function equals the asymptote?

**Tip:** When analyzing exponential functions, always check the base \( b \) to understand whether the function is increasing or decreasing.

Learn how to create exponential functions of the form f(x) = a * b^x + c that have a horizontal asymptote at y=5. Explore two different examples with detailed explanations and understand the role of constants a, b, and c in shaping the function's behavior. Suitable for high school students and anyone interested in understanding exponential functions.

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