https://file.aiquickdraw.com/mathbot/file/1728774496980icnfoc85.jpg

To match each equation to the appropriate graph, let’s analyze the equations and the general shapes of the graphs:

1. **Equation Analysis**:
   - **\( y = 1,000(0.95)^x \)**: This is an exponential decay function, since the base of the exponent (0.95) is less than 1. We expect a graph that starts high and decreases over time.
   - **\( y = 200 + 11x \)**: This is a linear function with a positive slope, indicating a straight line that increases as \( x \) increases.
   - **\( y = 5(1.1)^x \)**: This is an exponential growth function, since the base (1.1) is greater than 1. We expect a graph that starts low and increases sharply as \( x \) increases.

2. **Graph Analysis**:
   - **Graph A** shows an exponential growth shape: it starts low and increases rapidly. This graph matches the exponential growth function \( y = 5(1.1)^x \).
   - **Graph B** shows an exponential decay shape: it starts high and decreases over time. This graph matches the exponential decay function \( y = 1,000(0.95)^x \).
   - **Graph C** is a straight line with a positive slope, which corresponds to a linear equation. This graph matches \( y = 200 + 11x \).

3. **Matching Each Graph to an Equation**:
   - **Graph A**: \( y = 5(1.1)^x \)
   - **Graph B**: \( y = 1,000(0.95)^x \)
   - **Graph C**: \( y = 200 + 11x \)

Would you like more details on exponential functions or how to interpret these graphs?

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Here are five questions to expand on this topic:
1. What are the general characteristics of exponential decay and growth functions?
2. How can we determine if a function is linear or non-linear by looking at its equation?
3. How does changing the base in an exponential function affect the graph's shape?
4. What are some real-world examples of exponential growth and decay?
5. How can you tell the difference between exponential and linear functions just by observing a graph?

**Tip:** Exponential growth and decay functions are often used to model populations and radioactive decay, respectively. Recognizing their graphs can help you understand these phenomena better!

Select the letter of each graph next to the equation that best matches it.

This exercise focuses on matching three equations to their corresponding graphs: one representing exponential growth (y = 5(1.1)^x), one for exponential decay (y = 1000(0.95)^x), and one linear function (y = 200 + 11x). Learn how to identify each graph shape and its properties.

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