consider the function f(x) = ab^x, where a >0 and b>1. The graph of the function f(x)

The function \( f(x) = ab^x \) is an exponential function, where \( a > 0 \) and \( b > 1 \). Let's break down the behavior and characteristics of its graph:

### Key Characteristics of the Graph of \( f(x) = ab^x \):

1. **Shape**:
   - The graph is an upward-sloping curve that increases rapidly as \( x \) increases. Since \( b > 1 \), the function is exponentially increasing.

2. **Y-intercept**:
   - The y-intercept occurs where \( x = 0 \). Substituting \( x = 0 \) into the function, we get \( f(0) = ab^0 = a \). Thus, the y-intercept is \( (0, a) \).

3. **Behavior as \( x \) Increases**:
   - As \( x \) increases, \( f(x) \) increases rapidly because \( b^x \) grows exponentially. Therefore, the graph rises sharply to the right.

4. **Behavior as \( x \) Decreases**:
   - As \( x \) decreases (i.e., for negative values of \( x \)), \( f(x) \) approaches zero but never actually reaches it. This is because \( b^x \) becomes a fraction when \( x \) is negative, but since \( a > 0 \), \( f(x) \) remains positive.

5. **Horizontal Asymptote**:
   - The graph has a horizontal asymptote at \( y = 0 \). As \( x \) approaches negative infinity, \( f(x) \) gets closer and closer to zero but never actually touches the x-axis.

### Summary of the Graph's Behavior:
- The function's graph is an increasing exponential curve.
- The curve passes through the point \( (0, a) \), where \( a > 0 \).
- As \( x \) increases, the graph rises steeply.
- As \( x \) decreases, the graph approaches the x-axis but never touches it, indicating a horizontal asymptote at \( y = 0 \).

Would you like more details, or do you have any specific questions?

Here are five related questions to explore:

1. How does changing the value of \( a \) affect the graph of \( f(x) \)?
2. How does changing the value of \( b \) (while keeping \( b > 1 \)) affect the steepness of the graph?
3. What happens to the graph if \( b = 1 \)?
4. How does the graph change if \( 0 < b < 1 \)?
5. Can you describe the graph of \( f(x) = ab^{-x} \) and how it differs from \( f(x) = ab^x \)?

**Tip:** For any exponential function, the base \( b \) determines the rate of growth or decay. If \( b > 1 \), the function grows exponentially; if \( 0 < b < 1 \), it decays exponentially.

Explore the characteristics and behavior of the exponential function f(x) = ab^x. Learn about its upward-sloping curve, y-intercept at (0, a), rapid increase with increasing x, and horizontal asymptote at y = 0. Discover how changes in parameters a and b affect the graph's shape and steepness.

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