Let's go through the matching process step by step:

Functions:

• a. $y = 2^x$
• b. $y = 10^x$
• c. $y = -5^x$
• d. $y = \left( \frac{1}{3} \right)^x$
• e. $y = -\left( \frac{1}{2} \right)^x$

Graphs:

• A: An exponential function with rapid growth.
• B: A moderately increasing exponential function.
• C: A slow exponential decay.
• D: A reflection of an exponential decay, since it goes downwards.
• E: A reflection of a growth function, inverted downward.

Matching:

1. Graph A shows rapid exponential growth. This fits b. $y = 10^x$, as the base 10 grows faster than 2.
2. Graph B is also exponential growth but slower compared to graph A. This matches a. $y = 2^x$.
3. Graph C shows exponential decay. This matches d. $y = \left( \frac{1}{3} \right)^x$, as the base $\frac{1}{3}$ is less than 1.
4. Graph D is a reflection of an increasing function but downward, indicating a negative exponential decay. This matches e. $y = -\left( \frac{1}{2} \right)^x$.
5. Graph E is a reflection of a growth function going downward, which corresponds to c. $y = -5^x$.

Final Matching:

• a. $y = 2^x$ → B
• b. $y = 10^x$ → A
• c. $y = -5^x$ → E
• d. $y = \left( \frac{1}{3} \right)^x$ → C
• e. $y = -\left( \frac{1}{2} \right)^x$ → D

Would you like further details or clarifications?

Here are 5 related questions:

1. What is the general form of an exponential function?
2. How does the base of an exponential function affect its growth or decay?
3. What is the effect of a negative sign on the graph of an exponential function?
4. How can exponential functions model real-world growth and decay processes?
5. What are the key differences between exponential and linear functions?

Tip: When analyzing exponential graphs, always pay attention to the base and sign of the function to understand the rate and direction of growth or decay.