Q. Which is the graph of \(g(x)=\left(\tfrac{1}{2}\right)^{x+3}-4\)?

Problem: Analyze and graphically describe the function \( g(x)=\bigl(\tfrac{1}{2}\bigr)^{x+3}-4 \).

Step 1 — Parent function

The exponential base is \( \tfrac{1}{2} \), so the parent function is
\[
f(x)=\bigl(\tfrac{1}{2}\bigr)^{x},
\]
which is an exponential decay function (values decrease as \(x\) increases).

Step 2 — Horizontal shift

The exponent is \(x+3\). Write
\[
x+3=x-(-3),
\]
so the graph of the parent function is shifted 3 units to the left. Equivalently,
\[
g(x)=f(x+3)-4.
\]

Step 3 — Vertical shift and horizontal asymptote

The constant \(-4\) outside the exponential shifts the graph 4 units downward. The parent function has horizontal asymptote \(y=0\); after the vertical shift the asymptote becomes
\[
y=-4.
\]

Step 4 — y-intercept

Set \(x=0\):
\[
g(0)=\bigl(\tfrac{1}{2}\bigr)^{0+3}-4=\bigl(\tfrac{1}{2}\bigr)^{3}-4=\tfrac{1}{8}-4=-\tfrac{31}{8}=-3.875.
\]
Thus the y-intercept is \((0,-\tfrac{31}{8})\).

Step 5 — x-intercept

Set \(g(x)=0\) and solve:
\[
0=\bigl(\tfrac{1}{2}\bigr)^{x+3}-4
\quad\Rightarrow\quad
\bigl(\tfrac{1}{2}\bigr)^{x+3}=4.
\]
Write both sides with base \(2\):
\[
(2^{-1})^{x+3}=2^{2}\quad\Longrightarrow\quad 2^{-(x+3)}=2^{2}.
\]
Equating exponents gives
\[
-(x+3)=2 \quad\Rightarrow\quad x+3=-2 \quad\Rightarrow\quad x=-5.
\]
So the x-intercept is \((-5,0)\).

Summary

\[
g(x)=\bigl(\tfrac{1}{2}\bigr)^{x+3}-4
\]
is an exponential decay curve obtained from \(f(x)=\bigl(\tfrac{1}{2}\bigr)^{x}\) by shifting 3 units left and 4 units down. Its horizontal asymptote is \(y=-4\), the x-intercept is \((-5,0)\), and the y-intercept is \((0,-\tfrac{31}{8})\).