Which graph shows $$ f(x)=3 \cdot 2^{x}-2 $$ and its translation $$ g(x)=3 \cdot 2^{x}+3 ? $$

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1. We start with the function  
   $$
   f(x)=3\cdot 2^x-2.
   $$
   Its horizontal asymptote is determined by the constant term after the exponential part vanishes. As $$x\to -\infty$$,  
   $$
   3\cdot 2^x\to 0,
   $$
   so the horizontal asymptote is  
   $$
   y=-2.
   $$

2. Next, consider the translation given by  
   $$
   g(x)=3\cdot 2^x+3.
   $$
   Notice that compared to $$f(x)$$, the only change is that the constant $$-2$$ is replaced by $$+3$$. This is equivalent to shifting $$f(x)$$ vertically upward by 5 units (since $$3-(-2)=5$$). Therefore, $$g(x)$$ is the graph of $$f(x)$$ translated upward by 5.

3. The horizontal asymptote of $$g(x)$$ is determined similarly. As $$x\to -\infty$$,
   $$
   3\cdot 2^x\to 0,
   $$
   so the horizontal asymptote of $$g(x)$$ is  
   $$
   y=3.
   $$

4. To identify the graphs:
   - The graph of $$f(x)=3\cdot 2^x-2$$ is an exponential curve that approaches the line $$y=-2$$ as $$x\to -\infty$$ and increases rapidly as $$x\to\infty$$.
   - The graph of $$g(x)=3\cdot 2^x+3$$ is an exponential curve with the same shape as $$f(x)$$ but shifted upward by 5 units. Its horizontal asymptote is $$y=3$$.

5. Additionally, if we check a couple of key points:
   - For $$f(x)$$:  
     $$
     f(0)=3\cdot 2^0-2=3-2=1,
     $$  
     $$
     f(1)=3\cdot 2^1-2=6-2=4.
     $$
   - For $$g(x)$$:  
     $$
     g(0)=3\cdot 2^0+3=3+3=6,
     $$  
     $$
     g(1)=3\cdot 2^1+3=6+3=9.
     $$

6. Therefore, the correct graph will display:
   - Two exponential curves with the same shape.
   - One curve with a horizontal asymptote at $$y=-2$$ going through $$(0,1)$$ and $$(1,4)$$ for $$f(x)$$.
   - The other curve, which is just $$f(x)$$ shifted upward by 5, having a horizontal asymptote at $$y=3$$ and passing through $$(0,6)$$ and $$(1,9)$$ for $$g(x)$$.

The graph that shows these features—exponential curves with the described key points and asymptotes—is the one that correctly represents the functions $$f(x)=3\cdot 2^x-2$$ and its vertical translation $$g(x)=3\cdot 2^x+3$$.
The graph shows two exponential curves. One has a horizontal asymptote at $$ y = -2 $$ and passes through the points $$(0,1)$$ and $$(1,4)$$. The other curve is the first one shifted upward by 5 units, with a horizontal asymptote at $$ y = 3 $$ and passing through $$(0,6)$$ and $$(1,9)$$.

Which graph shows \( f(x)=3 \cdot 2^{x}-2 \) and its translation \( g(x)=3 \cdot 2^{x}+3 ? \)

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