This question involves the transformation of the exponential function $f(x) = 2^x$ to the given function $h(x) = 2^{x-1} - 2$.

Explanation:

1. Horizontal Shift:

   • The term $x-1$ inside the exponent indicates a horizontal shift of 1 unit to the right.

2. Vertical Shift:

   • The term $-2$ outside the function indicates a vertical shift of 2 units downward.

Thus, the correct transformation is:

• Horizontal shift 1 unit to the right.
• Vertical shift 2 units downward.

The correct answer is C: "The graph of $f(x) = 2^x$ should be horizontally shifted 1 unit to the right and vertically shifted 2 units downward."

Additional Details:

• The horizontal shift moves the asymptote from $y = 0$ to $y = -2$, because the function is shifted downward by 2 units.
• The domain of $h(x)$ remains $(-\infty, \infty)$, but the range changes to $(-2, \infty)$.

Let me know if you'd like further assistance with this problem or if you'd like to see the graph of this function!

Related Questions:

1. How does shifting an exponential function affect its asymptote?
2. What is the domain and range of the function $h(x) = 2^{x-1} - 2$?
3. How do transformations affect the rate of growth of an exponential function?
4. What happens to the graph of $f(x) = 2^x$ when it is reflected across the x-axis?
5. How can transformations be combined to model real-world exponential growth or decay?

Tip:

When analyzing transformations, always focus on changes to the base function step-by-step, identifying horizontal, vertical, and reflection transformations systematically.