To transform the graph of the function $f(x) = 2^x$ by first stretching it horizontally by a factor of 2 and then shifting it to the right by 5 units, we need to apply the transformations step by step.

Step 1: Stretching the graph horizontally by a factor of 2

To stretch the graph horizontally by a factor of 2, we modify the function by replacing $x$ with $\frac{x}{2}$. This means that the graph will become "wider."

The transformed function after horizontal stretching is: $f(x) = 2^{\frac{x}{2}}$

Step 2: Shifting the graph to the right by 5 units

To shift the graph to the right by 5 units, we replace $x$ with $x - 5$. This shifts the graph to the right by 5 units.

The final function after the rightward shift is: $h(x) = 2^{\frac{x - 5}{2}}$

Final Answer:

Thus, the equation of the function $h(x)$ is: $h(x) = 2^{\frac{x - 5}{2}}$

Would you like any further details or clarifications on this process?

Here are 5 related questions to explore:

1. How does a horizontal stretch affect the growth rate of exponential functions?
2. What happens to the graph of $2^x$ when you shift it to the left instead of to the right?
3. How would you stretch the graph vertically instead of horizontally?
4. What are some real-world examples where horizontal transformations of exponential functions are useful?
5. How would the function change if the horizontal stretch was by a factor of 3 instead of 2?

Tip: When applying multiple transformations to a function, always start with the innermost transformation and work outward. This ensures the correct order of operations!