Let's analyze the given function [tex]\( y = 3^{x+1} + 4 \)[/tex] to determine its horizontal shift in comparison to the original function [tex]\( y = 3^x \)[/tex].
1. Original Function:
[tex]\( y = 3^x \)[/tex]
2. Transformed Function:
[tex]\( y = 3^{x+1} + 4 \)[/tex]
We need to focus on the part of the transformed function that affects the horizontal shift. This component is [tex]\( 3^{x+1} \)[/tex].
3. Understanding the Transformation:
- The transformation [tex]\( 3^x \)[/tex] to [tex]\( 3^{x+1} \)[/tex] involves a horizontal shift.
- For exponential functions of the form [tex]\( 3^{x+c} \)[/tex], where [tex]\( c \)[/tex] is a constant, the graph shifts horizontally by [tex]\( -c \)[/tex] units.
- In this case, the constant [tex]\( c \)[/tex] is 1. This means the transformation inside the exponent is [tex]\( x \to x + 1 \)[/tex].
4. Determine the Direction and Magnitude of the Shift:
- A transformation of [tex]\( x \to x + 1 \)[/tex] indicates a horizontal shift to the left.
- The amount of the horizontal shift is given by the constant [tex]\( c \)[/tex].
5. Conclusion:
- The function [tex]\( y = 3^{x+1} \)[/tex] is shifted to the left by 1 unit relative to the function [tex]\( y = 3^x \)[/tex].
Therefore, the horizontal shift of the given function [tex]\( y = 3^{x+1} + 4 \)[/tex] relative to the original function [tex]\( y = 3^x \)[/tex] is [tex]\(-1\)[/tex] unit.
Horizontal shift: [tex]\(-1\)[/tex] unit
This means the graph of [tex]\( y = 3^{x+1} + 4 \)[/tex] is shifted to the left by 1 unit compared to [tex]\( y = 3^x \)[/tex].
