To transform the parent function [tex]\( f(x) = 5^x \)[/tex] based on the given transformations, follow these steps:
1. Vertical Compression by a Factor of One-half:
- To vertically compress the graph by a factor of [tex]\( \frac{1}{2} \)[/tex], multiply the function by [tex]\( \frac{1}{2} \)[/tex].
- This gives us: [tex]\( \frac{1}{2} \cdot 5^x \)[/tex].
2. Shift to the Right by Three Units:
- Shifting the function to the right by 3 units means replacing [tex]\( x \)[/tex] with [tex]\( (x - 3) \)[/tex].
- This modifies our function to: [tex]\( \frac{1}{2} \cdot 5^{(x - 3)} \)[/tex].
3. Shift Up by Two Units:
- To shift the graph up by 2 units, add 2 to the entire function.
- This results in the final transformed function: [tex]\( g(x) = \frac{1}{2} \cdot 5^{(x - 3)} + 2 \)[/tex].
Thus, the correct function representing all the mentioned transformations is:
[tex]\[ g(x) = \left( \frac{1}{2} \right) 5^{x - 3} + 2 \][/tex].
