which of the following describes the transformations of $g(x)=-(2)^{x + 4}-2$ from the parent function…

which of the following describes the transformations of $g(x)=-(2)^{x + 4}-2$ from the parent function $f(x)=2^{x}$?\nshift 4 units left, reflect over the x - axis, shift 2 units down\nshift 4 units left, reflect over the y - axis, shift 2 units down\nshift 4 units right, reflect over the x - axis, shift 2 units down\nshift 4 units right, reflect over the y - axis, shift 2 units down

which of the following describes the transformations of $g(x)=-(2)^{x + 4}-2$ from the parent function $f(x)=2^{x}$?\nshift 4 units left, reflect over the x - axis, shift 2 units down\nshift 4 units left, reflect over the y - axis, shift 2 units down\nshift 4 units right, reflect over the x - axis, shift 2 units down\nshift 4 units right, reflect over the y - axis, shift 2 units down

Answer

Explanation:

Step1: Analyze horizontal shift

For a function $y = f(x + h)$, if $h>0$, the graph shifts $h$ units to the left. In $g(x)=-(2)^{x + 4}-2$, compared to $f(x)=2^{x}$, we have $h = 4$, so it shifts 4 units left.

Step2: Analyze reflection

The negative sign in front of $(2)^{x+4}$ in $g(x)=-(2)^{x + 4}-2$ means $y = f(x)$ is reflected over the $x$-axis. When we have $y=-f(x)$, the graph of $y = f(x)$ is reflected across the $x$-axis.

Step3: Analyze vertical shift

For a function $y=f(x)+k$, if $k < 0$, the graph shifts $|k|$ units down. In $g(x)=-(2)^{x + 4}-2$, $k=-2$, so it shifts 2 units down.

Answer:

shift 4 units left, reflect over the x - axis, shift 2 units down