The given equation is $y = 3^{x-2}$, which represents an exponential function. Let's analyze it and describe how to sketch it.

Step-by-Step Analysis

1. Base Exponential Form:

   • The base is 3, meaning the function grows exponentially as $x$ increases.
   • The term $x - 2$ indicates a horizontal shift to the right by 2 units.

2. Key Features:

   • Asymptote: The graph approaches $y = 0$ but never touches it, as exponential functions have a horizontal asymptote at $y = 0$.
   • Intercept:
     • When $x = 2$: $y = 3^{2-2} = 3^0 = 1$, so the graph passes through $(2, 1)$.
   • Behavior:
     • For $x < 2$: The values decrease and approach 0 (but remain positive).
     • For $x > 2$: The values grow exponentially.

3. Key Points for the Graph: To sketch, calculate a few key points:

   • $x = 0$: $y = 3^{0-2} = 3^{-2} = \frac{1}{9}$.
   • $x = 1$: $y = 3^{1-2} = 3^{-1} = \frac{1}{3}$.
   • $x = 2$: $y = 1$.
   • $x = 3$: $y = 3^{3-2} = 3^1 = 3$.
   • $x = 4$: $y = 3^{4-2} = 3^2 = 9$.

4. Sketch:

   • Plot the points $(0, \frac{1}{9})$, $(1, \frac{1}{3})$, $(2, 1)$, $(3, 3)$, and $(4, 9)$.
   • Connect the points smoothly, ensuring the curve approaches $y = 0$ as $x \to -\infty$ and grows rapidly as $x \to \infty$.

Would you like me to generate an actual graph for you? Let me know!

Related Questions:

1. What is the effect of changing the base of the exponential function from 3 to another number, like 2 or 5?
2. How would the graph change if the equation were $y = 3^{-(x-2)}$?
3. What is the derivative of $y = 3^{x-2}$ and what does it tell us about the slope of the graph?
4. How does the horizontal shift $x - 2$ affect the graph compared to $y = 3^x$?
5. How can exponential functions model real-life phenomena like population growth or radioactive decay?

Tip:

To better understand exponential functions, try sketching their graphs with different bases and transformations (e.g., shifts, reflections, or scaling). It helps to visualize their behavior!