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.