https://file.aiquickdraw.com/mathbot/file/1729420765818his9jif9.jpg

The image shows four different graphs, each representing an exponential function. Based on the behavior of the curves, let's try to match each graph to its corresponding equation. Here's an analysis:

1. **First Graph (Dark blue curve)**: This is an increasing exponential curve passing through the point \((0, 1)\), which indicates it could be of the form \( y = a^x \) with \( a > 1 \). Given the steepness, this might be something like \( y = 10^x \).

2. **Second Graph (Orange curve)**: Another increasing exponential curve, but it grows even faster than the first one, suggesting that the base is larger. This could be a graph of \( y = e^x \), where \( e \) is approximately 2.718.

3. **Third Graph (Light blue curve)**: This graph shows a decreasing exponential curve. It seems to approach 0 as \( x \to \infty \), which suggests it's an exponential decay function. This might be a graph of \( y = e^{-x} \).

4. **Fourth Graph (Pink curve)**: Like the third one, this is also a decreasing exponential function. Since it decreases more gradually compared to the third, it could be a slower decay. This may represent something like \( y = 10^{-x} \).

Thus, a possible labeling could be:
- First graph: \( y = 10^x \)
- Second graph: \( y = e^x \)
- Third graph: \( y = e^{-x} \)
- Fourth graph: \( y = 10^{-x} \)

Would you like further details on why each graph corresponds to these equations? Here are 5 questions to further expand on this:

1. What is the general form of an exponential growth function?
2. How do you differentiate between exponential growth and decay on a graph?
3. Why does \( e^x \) grow faster than \( 10^x \) at larger values of \( x \)?
4. How do the signs of the exponents affect the shape of the graph?
5. What is the significance of the point \( (0, 1) \) for exponential functions?

**Tip**: The base of an exponential function determines how quickly the function grows or decays. Larger bases lead to steeper growth or decay curves.

Label each graph with the correct equation.

Learn how to identify and label exponential function graphs with equations like y = 10^x, y = e^x, and more. This problem covers key mathematical concepts of exponential growth and decay, ideal for high school students in Grades 9-12.

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