$$ f(x)=3-3^{x+1} $$ Parent Function: Transformations:

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To analyze the function $$ f(x) = 3 - 3^{x+1} $$, we can identify its parent function and the transformations applied to it.

### Parent Function
The parent function here is $$ g(x) = 3^x $$. This is an exponential function that has the following characteristics:
- It passes through the point $$ (0, 1) $$ since $$ 3^0 = 1 $$.
- It approaches $$ y = 0 $$ as $$ x $$ approaches negative infinity.
- It increases rapidly as $$ x $$ increases.

### Transformations
Now, let's break down the transformations applied to the parent function $$ g(x) = 3^x $$ to obtain $$ f(x) = 3 - 3^{x+1} $$:

1. **Horizontal Shift**: The term $$ x + 1 $$ indicates a horizontal shift to the left by 1 unit. This means that the graph of $$ g(x) = 3^x $$ is shifted left to become $$ g(x + 1) = 3^{x + 1} $$.

2. **Reflection**: The negative sign in front of $$ 3^{x + 1} $$ indicates a reflection across the x-axis. This transforms $$ g(x + 1) $$ into $$ -g(x + 1) = -3^{x + 1} $$.

3. **Vertical Shift**: Finally, the addition of 3 at the end of the function $$ -3^{x + 1} + 3 $$ indicates a vertical shift upwards by 3 units. This transforms $$ -g(x + 1) $$ into $$ f(x) = 3 - 3^{x + 1} $$.

### Summary of Transformations
- **Horizontal Shift**: Left by 1 unit
- **Reflection**: Across the x-axis
- **Vertical Shift**: Upward by 3 units

Thus, the complete transformation from the parent function $$ g(x) = 3^x $$ to $$ f(x) = 3 - 3^{x+1} $$ can be summarized as follows:

1. Shift left by 1: $$ g(x) 	o g(x + 1) $$
2. Reflect across the x-axis: $$ g(x + 1) 	o -g(x + 1) $$
3. Shift up by 3: $$ -g(x + 1) 	o f(x) = 3 - 3^{x + 1} $$
The function $$ f(x) = 3 - 3^{x+1} $$ is obtained from the parent function $$ g(x) = 3^x $$ by shifting left by 1 unit, reflecting across the x-axis, and shifting up by 3 units.

Parent Function:
Transformations:

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