match each exponential function to its graph.\n$u(x) = \\frac{1}{5}(3)^x$\n$v(x) = 3(3)^x$\n$w(x) =…

match each exponential function to its graph.\n$u(x) = \\frac{1}{5}(3)^x$\n$v(x) = 3(3)^x$\n$w(x) = 5(3)^x$\n$r(x) = 3\\left(\\frac{1}{5}\\right)^x$

match each exponential function to its graph.\n$u(x) = \\frac{1}{5}(3)^x$\n$v(x) = 3(3)^x$\n$w(x) = 5(3)^x$\n$r(x) = 3\\left(\\frac{1}{5}\\right)^x$

Answer

Explanation:

Step1: Find y-intercepts (x=0)

For $u(x)=\frac{1}{5}(3)^x$: $u(0)=\frac{1}{5}(3)^0=\frac{1}{5}=0.2$ For $v(x)=3(3)^x$: $v(0)=3(3)^0=3$ For $w(x)=5(3)^x$: $w(0)=5(3)^0=5$ For $r(x)=3\left(\frac{1}{5}\right)^x$: $r(0)=3\left(\frac{1}{5}\right)^0=3$

Step2: Classify growth/decay

  • Growth ($b>1$): $u(x), v(x), w(x)$ (increasing graphs)
  • Decay ($0<b<1$): $r(x)$ (decreasing graph)

Step3: Match decay function

Only decreasing graph (bottom left) has y-intercept 3, so $r(x)=3\left(\frac{1}{5}\right)^x$ → bottom left graph.

Step4: Match growth by y-intercept

  • Top left graph has y-intercept 3: $v(x)=3(3)^x$
  • Top right graph has y-intercept 5: $w(x)=5(3)^x$
  • Bottom right graph has y-intercept 0.2: $u(x)=\frac{1}{5}(3)^x$

Answer:

  • Top left graph: $v(x) = 3(3)^x$
  • Top right graph: $w(x) = 5(3)^x$
  • Bottom left graph: $r(x) = 3\left(\frac{1}{5}\right)^x$
  • Bottom right graph: $u(x) = \frac{1}{5}(3)^x$