Sure! Let's analyze the given function and determine the horizontal shift. The original function provided is:
[tex]\[ y = 3^x \][/tex]
We need to compare this to the transformed function provided:
[tex]\[ y = 3^{x+4} - 2 \][/tex]
To find out the horizontal shift, we focus on the exponent part of the transformed function relative to the original. Specifically, in the transformed function:
[tex]\[ 3^{x+4} \][/tex]
The term [tex]\( x + 4 \)[/tex] indicates that we are shifting the [tex]\( x \)[/tex] value within the exponent. When we have something of the form [tex]\( x + c \)[/tex] inside the exponent, this translates to a horizontal shift.
The rule for horizontal shifts is that:
- If the term inside the exponent is [tex]\( x + c \)[/tex], it indicates a shift to the left by [tex]\( c \)[/tex] units.
- If the term inside the exponent is [tex]\( x - c \)[/tex], it indicates a shift to the right by [tex]\( c \)[/tex] units.
In our case, the transformed function has [tex]\( x + 4 \)[/tex] in the exponent. Thus, [tex]\( +4 \)[/tex] means a shift to the left by 4 units.
Therefore, the direction and the amount of the horizontal shift for the function [tex]\( y = 3^{x+4} - 2 \)[/tex] from the original function [tex]\( y = 3^x \)[/tex] is:
Horizontal shift: [tex]\(-4\)[/tex] units (shifted 4 units to the left).
