Given the function f(X) is of the form f(X)=a*b^x+c , has a horizontal asymptote at y=-1, and passes through the point (0,2), create a possible function for f(X).

To create a function \( f(x) = a \cdot b^x + c \) that has a horizontal asymptote at \( y = -1 \) and passes through the point \( (0, 2) \), we follow these steps:

### Step 1: Identify the horizontal asymptote
The horizontal asymptote is at \( y = -1 \), which means \( c = -1 \) in the equation. So, the function becomes:
\[
f(x) = a \cdot b^x - 1
\]

### Step 2: Use the point \( (0, 2) \)
The function passes through the point \( (0, 2) \), which means when \( x = 0 \), \( f(x) = 2 \). Substituting \( x = 0 \) and \( f(0) = 2 \) into the equation:

\[
2 = a \cdot b^0 - 1
\]
Since \( b^0 = 1 \), the equation simplifies to:
\[
2 = a - 1
\]
Solving for \( a \):
\[
a = 3
\]

### Step 3: Final function
We now have \( a = 3 \) and \( c = -1 \), so the function becomes:
\[
f(x) = 3 \cdot b^x - 1
\]

### Step 4: Choose a base \( b \)
To complete the function, we need to choose a reasonable base \( b \). Let's assume \( b = 2 \) (which represents exponential growth).

Thus, a possible function is:
\[
f(x) = 3 \cdot 2^x - 1
\]

This function has a horizontal asymptote at \( y = -1 \) and passes through the point \( (0, 2) \).

Would you like more details or have any other questions?

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**Related questions:**
1. How do we verify if a point lies on a given exponential function?
2. What role does the value of \( b \) play in shaping the curve of the function?
3. Can you find other points on the graph of this function?
4. How would the graph change if \( a \) were negative?
5. What happens to the function as \( x \to \infty \) or \( x \to -\infty \)?

**Tip:** The point \( (0, f(0)) \) always gives you direct information about the parameter \( a + c \) in exponential functions, since \( b^0 = 1 \).

Learn how to create an exponential function f(x) = a * b^x + c with a horizontal asymptote at y = -1 and passing through the point (0, 2). This problem involves identifying the horizontal asymptote, using point-slope form, and choosing appropriate parameters a, b, and c. Suitable for students in Grades 10-12.

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