From the shape of the curve shown in the graph, the function $f(x) = a \cdot b^x$ is exponential and decaying, meaning the graph is decreasing as $x$ increases.

For exponential decay to occur, the base $b$ must satisfy $0 < b < 1$. Let's analyze the possible values of $b$ provided:

• $\frac{18}{5} = 3.6$: This is greater than 1, so it would represent exponential growth, not decay. This value is not valid.
• $\frac{1}{10} = 0.1$: This is between 0 and 1, so it represents exponential decay. This value is valid.
• $\frac{9}{10} = 0.9$: This is also between 0 and 1, representing exponential decay. This value is valid.
• $1.3$: This is greater than 1, representing exponential growth, so it is not valid for this decaying curve.
• $0.3$: This is between 0 and 1, representing exponential decay. This value is valid.

Possible values of $b$: $\frac{1}{10}, \frac{9}{10}, 0.3$

Would you like further clarification or more details on exponential functions?

5 Related Questions:

1. What conditions determine if an exponential function represents growth or decay?
2. How does the value of $a$ (the coefficient) affect the graph of an exponential function?
3. How can the rate of decay be interpreted from the base $b$ of an exponential function?
4. What happens to the graph when $b = 1$ in an exponential function?
5. How do transformations such as vertical shifts or reflections affect the graph of exponential functions?

Tip:

For exponential decay, smaller values of $b$ (closer to 0) lead to faster decay, while values closer to 1 result in slower decay.