Question

which equation is represented by the graph below?
answer
\\( y = 2^{-x - 2} \\)
\\( y = -2^{-x} - 2 \\)
\\( y = -2^{-x - 2} \\)
\\( y = 2^{-x} - 2 \\)
submit answer

Explanation:

Step1: Find y-intercept (x=0)

Test each option at $x=0$:

• Option1: $y=2^{-0-2}=2^{-2}=\frac{1}{4}=0.25$
• Option2: $y=-2^{-0}-2=-1-2=-3$
• Option3: $y=-2^{-0-2}=-2^{-2}=-\frac{1}{4}=-0.25$
• Option4: $y=2^{-0}-2=1-2=-1$

Step2: Match graph's y-intercept

The graph crosses y-axis at $y=-2$. Wait, recheck Option2: $y=-2^{-x}-2$, when $x\to+\infty$, $2^{-x}\to0$, so $y\to-2$, which matches the horizontal asymptote. When $x=0$, $y=-1-2=-3$, which matches the graph's point at x=0. When $x$ decreases (goes negative), $-x$ becomes positive, $2^{-x}$ grows, so $y=-2^{-x}-2$ decreases to $-\infty$, which matches the graph's leftward trend.

Answer:

B. $y = -2^{-x} - 2$