To determine the transformations applied to the function [tex]\( f(x) = 2^x \)[/tex] to obtain [tex]\( g(x) = \frac{1}{3} \cdot 2^x - 7 \)[/tex], let's analyze each component of the transformed function step-by-step:
1. Transformation involving [tex]\( \frac{1}{3} \cdot 2^x \)[/tex]:
- The multiplicative factor [tex]\( \frac{1}{3} \)[/tex] influences the graph of the function.
- Multiplying the function by [tex]\( \frac{1}{3} \)[/tex] compresses the graph vertically because the value of [tex]\( \frac{1}{3} \)[/tex] is between 0 and 1 (but not equal to 0). Therefore, the graph is pressed closer to the x-axis.
2. Transformation involving [tex]\( - 7 \)[/tex]:
- The subtraction of 7 from the function's expression impacts the vertical position of the graph.
- Subtracting 7 will move the entire graph downward by 7 units.
Given these transformations, we identify two specific changes:
- Vertical Compression: The factor of [tex]\( \frac{1}{3} \)[/tex] causes a vertical compression of the graph.
- Vertical Shift Down: The subtraction of [tex]\( 7 \)[/tex] shifts the graph downward by 7 units.
Thus, the correct selections are:
- E. It is vertically compressed.
- A. It is shifted down.
So, the graph of [tex]\( f(x) = 2^x \)[/tex] undergoes these two transformations to become [tex]\( g(x) = \frac{1}{3} \cdot 2^x - 7 \)[/tex]. The final answer is:
A. It is shifted down.
E. It is vertically compressed.
