Question

Graph the function.

\[f(x)=6\cdot\left(\dfrac{1}{3}\right)^x\]

Answer 1

f(x)=6*(1/3)^x is an exponential decay with base 1/3 and initial value 6.

Key features
- Domain: (-∞, ∞)
- Range: (0, ∞)
- y‑intercept: f(0)=6
- No x‑intercept (never crosses y=0)
- Horizontal asymptote: y=0
- Monotonic: strictly decreasing (as x increases, f(x)→0; as x→−∞, f(x)→∞)

Table of values (use these to plot)
- x = −2 → f(−2) = 6·3^2 = 54
- x = −1 → f(−1) = 6·3 = 18
- x = 0 → f(0) = 6
- x = 1 → f(1) = 2
- x = 2 → f(2) = 2/3 ≈ 0.667
- x = 3 → f(3) = 2/9 ≈ 0.222

Graphing instructions
1. Draw axes and the horizontal asymptote y=0.
2. Plot the points from the table (especially (0,6), (1,2), (2,2/3)).
3. Connect them with a smooth curve that falls rapidly to the right and rises without bound to the left, approaching the x‑axis but never touching it.

(If you want, I can produce an actual plotted image of this graph.)