what is the graph of the function $f(x)=\frac{1}{3}(6^{x})$?
what is the graph of the function $f(x)=\frac{1}{3}(6^{x})$?
Answer
Answer:
The graph of the function (y = \frac{1}{3}(6^{x})) is an exponential - growth curve. It passes through the point ((0,\frac{1}{3})) (since when (x = 0), (y=\frac{1}{3}(6^{0})=\frac{1}{3}(1)=\frac{1}{3})). As (x\to-\infty), (y\to0) (the (x) - axis is a horizontal asymptote), and as (x\to+\infty), (y) increases without bound.
Explanation:
Step1: Find the y - intercept
Set (x = 0). (y=\frac{1}{3}(6^{0})) Since any non - zero number to the power of (0) is (1), (y=\frac{1}{3}(1)=\frac{1}{3}).
Step2: Analyze the end - behavior as (x\to-\infty)
As (x\to-\infty), (6^{x}=\frac{1}{6^{-x}}). As (-x\to+\infty), (6^{-x}\to+\infty), so (\frac{1}{6^{-x}}\to0). Then (y = \frac{1}{3}(6^{x})\to0).
Step3: Analyze the end - behavior as (x\to+\infty)
As (x\to+\infty), (6^{x}) increases without bound. Since (y=\frac{1}{3}(6^{x})), (y) also increases without bound.