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 exponential functions of the form \( f(x) = a \cdot b^x + c \), with a horizontal asymptote at \( y = 5 \), we need the constant \( c \) in the equation to equal 5. This is because the horizontal asymptote in exponential functions is determined by the constant term \( c \).

### Function 1:
Let \( a = 3 \), \( b = 2 \), and \( c = 5 \).

So, the first function is:
\[
f(x) = 3 \cdot 2^x + 5
\]
This function has a horizontal asymptote at \( y = 5 \).

### Function 2:
Let \( a = -2 \), \( b = 0.5 \), and \( c = 5 \).

So, the second function is:
\[
f(x) = -2 \cdot (0.5)^x + 5
\]
This function also has a horizontal asymptote at \( y = 5 \).

Both functions approach \( y = 5 \) as \( x \to \infty \) or \( x \to -\infty \), depending on the growth or decay rate.

Would you like further details or have any questions?

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**Related questions:**
1. How do we find the horizontal asymptote of an exponential function in general?
2. What effect does changing \( a \) have on the graph of the exponential function?
3. How does the base \( b \) influence the growth or decay of the function?
4. Can exponential functions cross their horizontal asymptote?
5. How do vertical and horizontal shifts affect the asymptote of exponential functions?

**Tip:** In an exponential function, the base \( b \) determines whether the function represents growth (\( b > 1 \)) or decay (\( 0 < b < 1 \)).

Learn how to create two exponential functions of the form f(x) = a * b^x + c with a horizontal asymptote at y=5. Detailed examples with different values of parameters a, b, and c provided.

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