which is a shrink of an exponential growth function?\n○ $f(x) = \\frac{1}{3}(3)^x$\n○ $f(x) = 3(3)^x$\n○…
which is a shrink of an exponential growth function?\n○ $f(x) = \\frac{1}{3}(3)^x$\n○ $f(x) = 3(3)^x$\n○ $f(x) = \\frac{1}{3}\\left(\\frac{1}{3}\\right)^x$\n○ $f(x) = 3\\left(\\frac{1}{3}\\right)^x$
Answer
Explanation:
Step1: Recall exponential function form
The general form of an exponential function is ( f(x) = a(b)^x ), where:
- If ( b > 1 ), it's an exponential growth function.
- If ( 0 < b < 1 ), it's an exponential decay function.
- A vertical shrink occurs when ( 0 < a < 1 ), and a vertical stretch when ( a > 1 ).
Step2: Analyze each option
-
Option 1: ( f(x)=\frac{1}{3}(3)^x )
Here, ( b = 3 > 1 ) (growth), and ( a=\frac{1}{3} ) (since ( 0 < \frac{1}{3} < 1 ), this is a vertical shrink of the growth function ( y = 3^x )). -
Option 2: ( f(x)=3(3)^x )
( a = 3 > 1 ), so this is a vertical stretch of ( y = 3^x ), not a shrink. -
Option 3: ( f(x)=\frac{1}{3}\left(\frac{1}{3}\right)^x )
( b=\frac{1}{3} < 1 ), so this is an exponential decay function, not growth. -
Option 4: ( f(x)=3\left(\frac{1}{3}\right)^x )
( b=\frac{1}{3} < 1 ), so this is an exponential decay function, not growth.