To graph the function [tex]\( f(x) = 3 + 3^{x+3} \)[/tex], you need to apply a series of transformations to the parent function [tex]\( g(x) = 3^x \)[/tex]. Let's go through this step-by-step:
### Step-by-Step Transformation:
1. Horizontal Shift:
- The term [tex]\( x+3 \)[/tex] inside the exponent indicates a horizontal shift.
- Specifically, [tex]\( x+3 \)[/tex] shifts the graph 3 units to the left.
2. Vertical Shift:
- The term [tex]\( +3 \)[/tex] outside the exponential function indicates a vertical shift.
- This moves the graph 3 units up.
With these transformations applied, here is what we get:
### Horizontal Shift:
The function [tex]\( 3^{x+3} \)[/tex] is obtained by shifting [tex]\( 3^x \)[/tex] to the left by 3 units.
### Vertical Shift:
To shift the resulting graph up by 3 units, we get:
[tex]\[ f(x) = 3 + 3^{x+3} \][/tex]
### Domain:
The domain of [tex]\( f(x) = 3 + 3^{x+3} \)[/tex] is all real numbers, since exponential functions are defined for all real values of [tex]\( x \)[/tex].
### Range:
The range of the original function [tex]\( g(x) = 3^x \)[/tex] is [tex]\( (0, \infty) \)[/tex]. After shifting the graph up by 3 units, the new range becomes [tex]\( (3, \infty) \)[/tex].
### Horizontal Asymptote:
The horizontal asymptote of [tex]\( g(x) = 3^x \)[/tex] is [tex]\( y = 0 \)[/tex]. After shifting the graph up by 3 units, the new horizontal asymptote becomes [tex]\( y = 3 \)[/tex].
### Summary of Transformations:
Identifying the transformations:
- Horizontal shift left by 3 units.
- Vertical shift up by 3 units.
### Answer Selection:
To obtain the graph of [tex]\( f(x) = 3 + 3^{x+3} \)[/tex], we apply the following transformations:
- [tex]\( D \)[/tex] Horizontal shift: Shift 3 units to the left.
- [tex]\( 14 \)[/tex] Vertical shift: Shift 3 units up.
Thus, the final transformations needed are:
- D. Horizontal shift
- 14. Vertical shift
