1. **Stating the problem:** We are given the function $$y = \left(\frac{1}{3}\right)^x$$ and want to understand its behavior and graph. 2. **Formula and rules:** The function is an exponential function of the form $$y = a^x$$ where $$a = \frac{1}{3}$$. - Since $$0 < a < 1$$, this is an exponential decay function. - The graph decreases as $$x$$ increases. - The horizontal asymptote is $$y = 0$$ because $$a^x > 0$$ for all real $$x$$. 3. **Intermediate work and explanation:** - At $$x=0$$, $$y = \left(\frac{1}{3}\right)^0 = 1$$. - At $$x=1$$, $$y = \frac{1}{3}$$. - At $$x=-1$$, $$y = \left(\frac{1}{3}\right)^{-1} = 3$$. 4. **Summary:** - The function decreases from left to right. - It passes through the point $$(0,1)$$. - It approaches zero but never touches the x-axis. **Final answer:** The function $$y = \left(\frac{1}{3}\right)^x$$ is an exponential decay function with horizontal asymptote $$y=0$$ and passes through $$(0,1)$$.