reflection graph of y=3^x +2 with respect to line y=x

To reflect the graph of the function \( y = 3^x + 2 \) with respect to the line \( y = x \), we need to find the inverse of the function \( y = 3^x + 2 \), as reflecting over the line \( y = x \) swaps the roles of \( x \) and \( y \).

### Steps:

1. **Start with the original equation**:  
   \[ y = 3^x + 2 \]

2. **Swap \( x \) and \( y \) to reflect over the line \( y = x \)**:  
   \[ x = 3^y + 2 \]

3. **Solve for \( y \)** (which will now be the equation of the reflected function):
   \[
   x - 2 = 3^y
   \]
   Take the natural logarithm of both sides:
   \[
   \ln(x - 2) = \ln(3^y)
   \]
   Using logarithmic properties:
   \[
   \ln(x - 2) = y \ln(3)
   \]
   Solve for \( y \):
   \[
   y = \frac{\ln(x - 2)}{\ln(3)}
   \]

Thus, the equation of the reflected graph is:
\[
y = \frac{\ln(x - 2)}{\ln(3)}
\]

### Final Answer:
The reflection of \( y = 3^x + 2 \) with respect to the line \( y = x \) is:
\[
y = \frac{\ln(x - 2)}{\ln(3)}
\]

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Would you like a detailed explanation of any part of this process? Here are five related questions to expand your understanding:

1. How do we find the inverse of exponential functions in general?
2. What are the properties of logarithms that help solve such equations?
3. How does reflecting a function over the line \( y = x \) relate to finding its inverse?
4. What does the graph of \( y = 3^x + 2 \) look like before and after reflection?
5. How do vertical and horizontal shifts affect reflections over the line \( y = x \)?

**Tip**: Reflecting over the line \( y = x \) is the same as finding the inverse of a function. Always remember to swap \( x \) and \( y \) and then solve for \( y \).

Explore how to find the reflection of the exponential function y=3^x + 2 over the line y=x. This step-by-step guide walks through inverse functions and logarithmic transformations, suitable for Grades 10-12 students.

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