Question

How do you determine if the equation \[y={{\left( \dfrac{1}{4} \right)}^{x}}\] represent exponential growth or decay.

Answer 1

Hint: In this problem, we have to determine if the equation represents exponential growth or decay. we know that any number divided by 1, always results less than 1. We can now put the increasing values for x, we know that any number to the power 1 is 1 itself, we can substitute the increasing power values in denominator one by one. As if the exponent in the denominator increases the total result keeps on decreasing.

Complete answer:
We know that the given equation is,
\[y={{\left( \dfrac{1}{4} \right)}^{x}}\]
We know that the exponent rule is
\[y={{\left( \dfrac{a}{b} \right)}^{x}}=\left( \dfrac{{{a}^{x}}}{{{b}^{x}}} \right)\] .
Now we can write the given equation as,
\[y=\left( \dfrac{1}{{{4}^{x}}} \right)\]
We also know that, anything to the power one is one itself.
Now we can substitute the values for x.
We can assume that x = 1, 2, 3, 4….
We can substitute the above values one by one, we get
\[\Rightarrow y=\left( \dfrac{1}{{{4}^{1}}} \right),\left( \dfrac{1}{{{4}^{2}}} \right),\left( \dfrac{1}{{{4}^{3}}} \right),\left( \dfrac{1}{{{4}^{4}}} \right),....\]
\[\begin{align}
  & \Rightarrow y=\dfrac{1}{4},\dfrac{1}{16},\dfrac{1}{64},\dfrac{1}{256} \\
 & \Rightarrow y=0.25,0.0625,0.0156,0.0039.... \\
\end{align}\]
Here, in this above step we can clearly see that, if the denominator, i.e., the exponent increases, the total value decreases.
We can conclude that, if x increases, \[{{\left( \dfrac{1}{4} \right)}^{x}}\] decreases.
Therefore, the given equation \[y={{\left( \dfrac{1}{4} \right)}^{x}}\] represent exponential decay.

Note: Students make mistakes while converting the fraction form to decimal, in order to find the exact decreasing values, to check whether the given equation is increasing or decreasing. We should also know that, in a fraction, if the denominator increases, then the total value decreases.