This image contains a graph with multiple exponential functions in the form $y = ab^x$. Let's address the three questions based on the graph:

1. Which graph has the largest value for $b$?

   • The value of $b$ determines how rapidly the graph grows or decays as $x$ increases. Graphs with larger $b$ values grow faster. Based on the image, the purple graph ($K$) grows the fastest for positive $x$, so it has the largest value of $b$.

2. Which graph has the smallest value for $b$?

   • A small value of $b$ corresponds to slow growth or fast decay. The red graph ($R$) decreases the fastest as $x$ increases, so it has the smallest $b$ value.

3. Which graph has the largest value for $a$?

   • The $a$ value determines the initial height (intercept) of the graph when $x = 0$. The green graph ($G$) starts at the highest point at $x = 0$, so it has the largest $a$.

Final Answers:

• Largest $b$: Purple graph ($K$).
• Smallest $b$: Red graph ($R$).
• Largest $a$: Green graph ($G$).

Would you like further explanation of these concepts or their applications?

Here are 5 follow-up questions for deeper understanding:

1. What happens to the shape of the graph as $b$ approaches 1?
2. How does changing $a$ affect the graph of $y = ab^x$?
3. Can $b$ be negative in $y = ab^x$? What would that look like?
4. How would you determine the exact $a$ and $b$ values from such a graph?
5. What are some real-life phenomena modeled by exponential functions?

Tip: Exponential graphs are often used to model population growth, radioactive decay, and compound interest—understanding their behavior is key in many scientific fields!