sketch the graph of the function.\n$f(x)=2(4)^{x}$
sketch the graph of the function.\n$f(x)=2(4)^{x}$
Answer
Explanation:
Step1: Find the y - intercept
When (x = 0), (f(0)=2(4)^{0}). Since (a^{0}=1) ((a\neq0)), then (f(0)=2\times1 = 2). So the point ((0,2)) is on the graph.
Step2: Find another point
When (x = 1), (f(1)=2(4)^{1}=2\times4 = 8). So the point ((1,8)) is on the graph.
Step3: Analyze the behavior as (x\to-\infty)
As (x\to-\infty), (y = 2(4)^{x}\to0) (because (4^{x}=\frac{1}{4^{-x}}) and as (x\to-\infty), (4^{-x}\to+\infty)). The horizontal asymptote is (y = 0).
Step4: Analyze the behavior as (x\to+\infty)
As (x\to+\infty), (y = 2(4)^{x}\to+\infty) since the base (a = 4>1) and the function (y = a^{x}) is an exponential growth function.
Plot the points ((0,2)) and ((1,8)), draw a smooth curve approaching the horizontal asymptote (y = 0) as (x\to-\infty) and increasing without bound as (x\to+\infty).
Answer:
The graph of (y = 2(4)^{x}) has a (y) - intercept at ((0,2)), passes through ((1,8)), has a horizontal asymptote (y = 0), and is an increasing exponential curve.