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.