Question

which equation is represented by the graph below?answer\\(\boldsymbol{y = 3^{-x} - 4}\\)\\(\boldsymbol{y = 3^{-x} + 4}\\)\\(\boldsymbol{y = \left(\frac{1}{3}\
ight)^{-x} + 4}\\)\\(\boldsymbol{y = \left(\frac{1}{3}\
ight)^{-x} - 4}\\)

Explanation:

Step1: Identify y-intercept

Find $y$ when $x=0$. The graph crosses $y$-axis at $(0, -3)$.

Step2: Test each option at $x=0$

• For $y=3^{-x}-4$: $y=3^{0}-4=1-4=-3$
• For $y=3^{-x}+4$: $y=3^{0}+4=1+4=5$
• For $y=(\frac{1}{3})^{-x}+4$: $y=(\frac{1}{3})^{0}+4=1+4=5$
• For $y=(\frac{1}{3})^{-x}-4$: $y=(\frac{1}{3})^{0}-4=1-4=-3$

Step3: Check right-end behavior

As $x\to+\infty$, graph approaches a horizontal asymptote above $y=-4$.

• For $y=3^{-x}-4$: $3^{-x}=\frac{1}{3^x}\to0$, so $y\to0-4=-4$ (matches)
• For $y=(\frac{1}{3})^{-x}-4=3^x-4$: $3^x\to+\infty$, so $y\to+\infty$ (does not match)

Answer:

$y = 3^{-x} - 4$