In an exponential equation like $f(x) = 3^{(x - 2)} + 1$, each component affects the graph in specific ways. Here’s a breakdown of the components:

1. The Base (3): The base in an exponential function $b^x$ (where $b > 1$) determines the rate of growth or decay. Here, since the base is 3 (which is greater than 1), it signifies exponential growth. As $x$ increases, the graph will rise more quickly than if the base were a smaller number, like 2. If the base were between 0 and 1 (like 0.5), the graph would represent exponential decay.

2. The $x - 2$ term:

   • Horizontal Shift: The $(x - 2)$ inside the exponent affects the horizontal position of the graph. Specifically, subtracting 2 from $x$ shifts the graph to the right by 2 units. Generally, in $b^{(x - h)}$, $h$ shifts the graph $h$ units to the right if positive and to the left if negative.
   • Horizontal Stretch/Compression: If there were a coefficient in front of $x$ (e.g., $3^{2x}$), it would cause a horizontal compression or stretch. Since we don't have that here, the graph isn't stretched or compressed horizontally.

3. The $+1$ outside the exponent:

   • Vertical Shift: The $+1$ outside the exponential expression shifts the entire graph vertically. Since it's $+1$, the graph moves up by 1 unit. This affects the horizontal asymptote (the line the graph approaches but never reaches) as well. Typically, the horizontal asymptote of $b^x$ is $y = 0$, but here, it shifts to $y = 1$.
   • Vertical Stretch/Compression: If there were a coefficient multiplying the exponential part (e.g., $2 \cdot 3^{(x - 2)}$), it would stretch or compress the graph vertically. Without such a coefficient here, there’s no vertical stretch or compression.

Summary

In $f(x) = 3^{(x - 2)} + 1$:

• The base $3$ causes exponential growth.
• $x - 2$ shifts the graph 2 units to the right.
• $+1$ shifts the graph 1 unit up, changing the asymptote to $y = 1$.

Would you like more details on any specific transformations, or have further questions?

Follow-Up Questions

1. How does changing the base from 3 to 5 affect the graph?
2. What happens if we add a coefficient in front of $3^{(x - 2)}$?
3. How would a negative sign in the exponent, like $3^{-(x - 2)}$, change the graph?
4. What would happen to the horizontal asymptote if we changed $+1$ to $-3$?
5. How does exponential growth compare to linear growth visually on a graph?

Tip

When analyzing transformations, focus on one part of the function at a time to see its specific effect on the graph.