Question

the graph of the function g(x) and its base exponential function f(x) = 2^x are displayed. indicate the transformation(s) of f(x) needed to create g(x). note: only factors of 0.5 or 2 are used for y - axis dilations.
history (select a row to reveal)
base function f(x) = 2^x
choose a transformation to change the graph.
reflection
dilation
horizontal translation
vertical translation
choose a transformation.
edit the transforming function.
f(x) = 2^x
transformation form: y = ±a·f(b(x - c)) + d

Explanation:

Step1: Identify basic function key point

The basic function $f(x)=2^x$ passes through $(0,1)$ and $(1,2)$.

Step2: Identify transformed function key point

The target function $g(x)$ passes through $(0,-1)$ and $(1,-2)$.

Step3: Match to transformation form

Compare points: $g(x) = -1 \cdot 2^{1(x-0)} + 0$. This is a reflection over the x-axis.

Answer:

The transformation needed is a reflection over the x-axis, so the function is $g(x) = -2^x$.