https://file.aiquickdraw.com/mathbot/file/1726475819140e29f31yn.jpg

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?

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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.

Match each function with its graph: a) y = 2^x b) y = 10^x c) y = -5^x d) y = (1/3)^x e) y = -(1/2)^x with graphs labeled A, B, C, D, and E.

Learn how to match exponential functions like y = 2^x, y = 10^x, and more with their corresponding graphs. This problem explains key concepts such as growth, decay, and graph reflections. Suitable for students in Grades 9-12.

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