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.