Which of the following describes the transformation of g(x) = 3 * (2) ^ (- x) + 2 from the parent function f(x) = 2 ^ x * 7

Let's break down the given functions and analyze the transformation:

1. **Parent function**: \( f(x) = 2^x \cdot 7 \)
   - This is an exponential function where the base is 2, and it is multiplied by 7, meaning the parent function grows exponentially with base 2.

2. **Transformed function**: \( g(x) = 3 \cdot 2^{-x} + 2 \)
   - This is also an exponential function, but it has several modifications from the parent function.

### Step-by-step transformation:

1. **Base change**: 
   - The base of both functions remains 2, so there is no change in the base.

2. **Negative exponent**:
   - \( 2^{-x} \) in \( g(x) \) represents a reflection across the **y-axis**. The negative sign in the exponent means the growth direction is reversed from that of the parent function \( 2^x \).

3. **Vertical stretch**:
   - The coefficient of 3 in \( g(x) = 3 \cdot 2^{-x} + 2 \) indicates a **vertical stretch** by a factor of 3 compared to the parent function, which would have a stretch factor of 7.

4. **Vertical translation**:
   - The "+2" outside the exponential expression indicates a **vertical shift upwards** by 2 units.

### Summary of the transformation:
- Reflection across the y-axis due to the negative exponent.
- Vertical stretch by a factor of 3.
- Vertical translation upwards by 2 units.

Would you like more details or have any specific questions?

Here are 5 related questions:
1. How does a horizontal shift affect an exponential function?
2. What does a negative exponent indicate geometrically?
3. How does changing the coefficient in front of an exponential function affect its graph?
4. How would the graph look if we added a horizontal shift in \( g(x) \)?
5. What happens when you change the base of an exponential function?

**Tip**: Remember that a negative exponent reflects the graph across the y-axis.

Which of the following describes the transformation of g(x) = 3 * (2) ^ (- x) + 2 from the parent function f(x) = 2 ^ x * 7?

Explore the transformation of the exponential function g(x) = 3 * (2)^(-x) + 2 from its parent function f(x) = 2^x * 7. Learn about reflections, vertical stretches, and translations, with clear explanations for high school students in Grades 10-12.

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