which of the following describes the transformation of $g(x)=3(2)^{-x}+2$ from the parent function…

which of the following describes the transformation of $g(x)=3(2)^{-x}+2$ from the parent function $f(x)=2^{x}$?\nreflect across the x - axis, stretch the graph vertically by a factor of 3, shift 2 units up\nreflect across the y - axis, stretch the graph vertically by a factor of 2, shift 3 units up\nreflect across the x - axis, stretch the graph vertically by a factor of 2, shift 3 units up\nreflect across the y - axis, stretch the graph vertically by a factor of 3, shift 2 units up

which of the following describes the transformation of $g(x)=3(2)^{-x}+2$ from the parent function $f(x)=2^{x}$?\nreflect across the x - axis, stretch the graph vertically by a factor of 3, shift 2 units up\nreflect across the y - axis, stretch the graph vertically by a factor of 2, shift 3 units up\nreflect across the x - axis, stretch the graph vertically by a factor of 2, shift 3 units up\nreflect across the y - axis, stretch the graph vertically by a factor of 3, shift 2 units up

Answer

Explanation:

Step1: Analyze sign - change of x

For the parent function $f(x)=2^{x}$ and the new function $g(x)=3(2)^{-x}+2$, the $x$ in the exponent has changed sign. When $y = f(x)$ is transformed to $y = f(-x)$, the graph is reflected across the y - axis.

Step2: Analyze vertical stretch

The coefficient in front of the exponential term is 3. For a function $y = f(x)$ transformed to $y = a\cdot f(x)$ ($a>0$), when $a = 3$, the graph is stretched vertically by a factor of 3.

Step3: Analyze vertical shift

The constant term added at the end is 2. For a function $y = f(x)$ transformed to $y=f(x)+k$, when $k = 2$, the graph is shifted 2 units up.

Answer:

reflect across the y - axis, stretch the graph vertically by a factor of 3, shift 2 units up