Question

describe the transformation taking place from the graph of f(x)=2^x to graph 2 (dashed black line). (1 point) graph 2 is showing -f(x) graph 2 is showing -f(-x) graph 2 is showing f(-x + 4) graph 2 is showing f(-x)

Explanation:

Step1: Recall transformation rules

For a function $y = f(x)$, $-f(x)$ reflects the graph of $y = f(x)$ over the $x -$axis, $f(-x)$ reflects the graph of $y = f(x)$ over the $y -$axis, and $-f(-x)$ reflects the graph of $y = f(x)$ over both the $x -$axis and $y -$axis. Also, $f(-x + 4)=f(-(x - 4))$ is a horizontal shift to the right by 4 units along with a $y -$axis reflection.

Step2: Analyze Graph 2

The graph of $y = 2^{x}$ has a $y -$intercept at $(0,1)$ and increases as $x$ increases. Graph 2 has a $y -$intercept at $(0, - 1)$ and increases as $x$ decreases. This means the graph of $y = 2^{x}$ has been reflected over both the $x -$axis and $y -$axis. When we transform $f(x)=2^{x}$ to $-f(-x)=-2^{-x}$, we first reflect $y = 2^{x}$ over the $y -$axis to get $y = 2^{-x}$ and then reflect it over the $x -$axis to get $y=-2^{-x}$.

Answer:

Graph 2 is showing $-f(-x)$