https://file.aiquickdraw.com/mathbot/file/1730324392922mgn6smc0.jpg

https://file.aiquickdraw.com/mathbot/file/1730324392924dio1ootj.jpg

Let's examine and solve the questions from each part of the assignment.

### Page 1: Exponential Transformations

1. **Top Graph**:
   - The original function \( f(x) = 2^x \) (in blue) has been modified. 
   - The black graph seems to be vertically stretched.
   - Transformation: **Vertical Stretch**.
   - New function: It could be in the form \( g(x) = k \cdot 2^x \) for some \( k > 1 \). Observing the y-values would help find \( k \).

2. **Middle Graph**:
   - The black graph is a **horizontal shift** to the right compared to the blue graph.
   - Transformation: **Horizontal Shift to the Right**.
   - New function: \( g(x) = 2^{x - c} \) where \( c \) is positive.

3. **Bottom Graph**:
   - The black graph is a **reflection** of the blue graph across the y-axis.
   - Transformation: **Reflection across the y-axis**.
   - New function: \( g(x) = 2^{-x} \).

### Page 2: Exponential Growth Scenarios

1. **Investor Scenario**:
   - The transformation that represents a delayed start (investor waiting to start) could be a **horizontal shift** to the right.
   - This delay represents the function \( f(x - c) \) where \( c \) is the delay in years.

2. **House Value Growth**:
   - Given the values, calculate the growth rate \( b \).
   - Use the general exponential model: \( y = a \cdot b^x \).
   - Initial value \( a = 350,000 \).
   - Calculate \( b \) based on the rate of increase year over year.

Would you like detailed calculations for each part? Let me know if you have specific questions on any part, or need clarification.

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### Relative Questions to Expand Learning:
1. How do you determine the exact vertical stretch factor from a graph?
2. How does a horizontal shift affect the intercepts of an exponential function?
3. What real-life situations can be modeled by reflections in exponential functions?
4. How can you calculate the exact growth rate from a set of values?
5. How does a delayed start (horizontal shift) affect long-term investments?

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#### Tip:
For exponential functions, transformations are often easier to understand when rewritten in their modified forms, such as \( f(x - c) \) for shifts or \( k \cdot f(x) \) for stretches.

Identify transformations in exponential functions and find an exponential model for house value growth.

Explore exponential function transformations and find the exponential model for house value growth based on historical data. This solution includes vertical stretch, horizontal shift, and reflection transformations, with applications in financial growth scenarios.

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