which function represents a vertical stretch of an exponential function?\n f(x)=3left(\frac{1}{2}\right)^{x}\…

which function represents a vertical stretch of an exponential function?\n f(x)=3left(\frac{1}{2}\right)^{x}\n f(x)=\frac{1}{2}(3)^{x}\n f(x)=(3)^{2x}\n f(x)=3^{left(\frac{1}{2}x\right)}

which function represents a vertical stretch of an exponential function?\n f(x)=3left(\frac{1}{2}\right)^{x}\n f(x)=\frac{1}{2}(3)^{x}\n f(x)=(3)^{2x}\n f(x)=3^{left(\frac{1}{2}x\right)}

Answer

Explanation:

Step1: Recall vertical - stretch rule

For an exponential function $y = a\cdot b^{x}$, if $|a|> 1$, it represents a vertical stretch.

Step2: Analyze each option

  • Option 1: $f(x)=3\left(\frac{1}{2}\right)^{x}$, here $a = 3>1$, it is a vertical stretch of the exponential function $y=\left(\frac{1}{2}\right)^{x}$.
  • Option 2: $f(x)=\frac{1}{2}(3)^{x}$, since $a=\frac{1}{2}<1$, it is a vertical compression of the exponential function $y = 3^{x}$.
  • Option 3: $f(x)=(3)^{2x}=9^{x}$, this is a transformation of the base of the exponential function, not a vertical - stretch.
  • Option 4: $f(x)=3^{\frac{1}{2}x}$, this is a horizontal transformation of the exponential function $y = 3^{x}$, not a vertical - stretch.

Answer:

$f(x)=3\left(\frac{1}{2}\right)^{x}$