To graph the function $f(x) = 2^{x - 2}$, let's break down the process step-by-step:

Step-by-Step Explanation

1. Identify the Base Function: The function $f(x) = 2^{x - 2}$ is an exponential function. Its base, $2$, is greater than 1, meaning it will show exponential growth as $x$ increases.

2. Shift the Function: The term $x - 2$ in the exponent implies a horizontal shift to the right by 2 units. The base function $2^x$ is shifted right by 2 units, making $f(x) = 2^{x - 2}$.

3. Determine Key Points:

   • At $x = 2$: $f(2) = 2^{2-2} = 2^0 = 1$.
   • At $x = 3$: $f(3) = 2^{3-2} = 2^1 = 2$.
   • At $x = 4$: $f(4) = 2^{4-2} = 2^2 = 4$.
   • At $x = 1$: $f(1) = 2^{1-2} = 2^{-1} = 0.5$.
   • At $x = 0$: $f(0) = 2^{0-2} = 2^{-2} = 0.25$.

4. Plot the Points: Using the points we calculated, plot each on the graph:

   • $(2, 1)$
   • $(3, 2)$
   • $(4, 4)$
   • $(1, 0.5)$
   • $(0, 0.25)$

5. Draw the Curve: Connect these points with a smooth curve, making sure the graph approaches the x-axis as $x \to -\infty$ but never touches it (since the x-axis is an asymptote for exponential functions).

Would you like me to provide more detailed steps on graphing or discuss additional concepts related to exponential functions? Here are some follow-up questions that might interest you:

1. What is the effect of changing the base of the exponential function?
2. How does shifting the function vertically or horizontally impact the graph?
3. What are the characteristics of exponential growth vs. exponential decay?
4. How would the graph look if the exponent was negative, such as $f(x) = 2^{-(x - 2)}$?
5. How can we use exponential functions to model real-world scenarios?

Tip: When working with transformations of functions, remember that changes inside the exponent (like $x - 2$) shift the graph horizontally, while changes outside the exponent affect the vertical position.