# Tell whether the function represents exponential growth or exponential decay. Identify the percent increase or decrease. Then graph the function. f(x)=left(frac{1}{3}right)^{x}

Question
Exponential growth and decay
Tell whether the function represents exponential growth or exponential decay. Identify the percent increase or decrease. Then graph the function.
$$\displaystyle{f{{\left({x}\right)}}}={\left({\frac{{{1}}}{{{3}}}}\right)}^{{{x}}}$$

2020-10-24
Step 1
We are given the function:
$$\displaystyle{f{{\left({x}\right)}}}={\left({\frac{{{1}}}{{{3}}}}\right)}^{{{x}}}$$
Step 2
We identify the value of the base. The base, $$\displaystyle{\frac{{{1}}}{{{3}}}},$$ is greater than 0 and smaller than 1, the function represents exponential decay.
Step 3
We make a table of values:
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&{y}\backslash{h}{l}\in{e}-{2}&{9}\backslash{h}{l}\in{e}-{1}&{3}\backslash{h}{l}\in{e}{0}&{1}\backslash{h}{l}\in{e}{1}&{\frac{{{1}}}{{{3}}}}\backslash{h}{l}\in{e}{2}&{\frac{{{1}}}{{{9}}}}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Step 4
We plot the points from the table. Then draw a smoth curve from right to left, that begins just above the x-axis, passes throigh the plotted points and moves up to the left.

Step 5
The percent of decrease is:
$$\displaystyle{1}\ -\ {\frac{{{1}}}{{{3}}}}={\frac{{{2}}}{{{3}}}}={66}\%$$

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