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The image you uploaded contains text and a graph related to "Exponential Models." Let me extract the key information from the image and provide a detailed explanation.

### Extracted Content:

1. **Title:** Exponential Models
2. **Introduction:** 
   - Exponential functions have the form \( y = ab^x \), where \( a \) is a non-zero constant and \( b \) is a positive constant other than 1.
   - The function models how a quantity \( y \) grows or decays over time. 
   - The value of \( b \) determines the nature of the function: \( b > 1 \) indicates growth, while \( 0 < b < 1 \) indicates decay.

3. **Example Problem:**
   - For the function \( y = 2(3)^x \), let \( x \) take on values 0, 1, 2, and 3. 
   - The corresponding \( y \)-values are calculated and listed in a table.
   - A graph is shown with a curve that passes through the points generated by the table.

4. **Table:**
   - A table showing values of \( x \) and \( y \):
     - \( x = 0 \), \( y = 2 \)
     - \( x = 1 \), \( y = 6 \)
     - \( x = 2 \), \( y = 18 \)
     - \( x = 3 \), \( y = 54 \)
   
5. **Graph:**
   - A graph illustrating the exponential growth of the function \( y = 2(3)^x \).
   - The curve is steep, showing rapid increase as \( x \) increases.

### Explanation:

Exponential functions are fundamental in modeling situations where quantities grow or decay at a constant relative rate. The general form of an exponential function is \( y = ab^x \), where:
- \( a \) represents the initial amount (when \( x = 0 \)).
- \( b \) is the growth factor (if \( b > 1 \)) or decay factor (if \( 0 < b < 1 \)).
- \( x \) is the independent variable, typically representing time.

In the provided example \( y = 2(3)^x \), the function models exponential growth because the base \( b = 3 \) is greater than 1. For each unit increase in \( x \), the value of \( y \) is multiplied by 3. The table provided shows the computed values of \( y \) for different values of \( x \), and the corresponding graph depicts how rapidly \( y \) increases as \( x \) increases.

### Questions and Further Exploration:

1. What happens to the graph of \( y = ab^x \) if the value of \( b \) is between 0 and 1?
2. How does the value of \( a \) affect the starting point of the graph?
3. Can you identify real-world phenomena that are modeled by exponential growth or decay?
4. How would the graph change if \( y = 2(3)^{-x} \) was plotted instead?
5. What are the characteristics of a logarithmic function, and how does it relate to exponential functions?

### Tip:
When plotting exponential functions, carefully choose a range of \( x \)-values that showcases the behavior of the function, especially around \( x = 0 \), to accurately represent its growth or decay. 

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Explain the exponential model using the function y = 2(3)^x.

Explore the exponential model using the function y = 2(3)^x. This lesson includes a detailed explanation, example problems, and a graph illustrating exponential growth, suitable for students in Grades 9-12.

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