Interpret the function [tex]f(x)=6 \cdot 5^{-2 x}[/tex] using the properties of exponents.
Drag and drop the correct words or values to complete the sentences.
The expression [tex]6 \cdot 5^{-2 x}[/tex] can be rewritten as: [ ]
The function [ ] by a factor of [ ] for each unit increase in [tex]x[/tex].
:: [tex]6.25^2[/tex]
:: 6
:: decays
:: [tex]\frac{1}{25}[/tex]
:: grows
:: [tex]6\left(\frac{1}{25}\right)^x[/tex]
Answer (2)
The expression 6 ⋅ 5 − 2 x can be rewritten as 6 ( 25 1 ) x . The function decays by a factor of 25 1 for each unit increase in x .
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The expression 6"." 5 − 2 x can be rewritten as 6 ( 25 1 ) x . The function decays by a factor of 25 1 for each unit increase in x .
Explanation
Understanding the Function We are given the function f ( x ) = 6"." 5 − 2 x and we want to rewrite it and describe its behavior.
Rewriting the Exponent First, we rewrite the expression 5 − 2 x using the property of exponents that states a b c = ( a b ) c . Therefore, we can rewrite 5 − 2 x as ( 5 − 2 ) x .
Simplifying the Base Next, we simplify 5 − 2 . Recall that a − n = a n 1 . Thus, 5 − 2 = 5 2 1 = 25 1 .
Rewriting the Function Now we can rewrite the function as f ( x ) = 6"." ( 25 1 ) x .
Determining Growth or Decay To determine if the function grows or decays, we look at the base of the exponential term. Since the base is 25 1 , which is between 0 and 1, the function decays.
Determining the Decay Factor The function decays by a factor of 25 1 for each unit increase in x . This means that for every increase of 1 in x , the value of the function is multiplied by 25 1 .
Final Answer Therefore, the expression 6"." 5 − 2 x can be rewritten as 6 ( 25 1 ) x . The function decays by a factor of 25 1 for each unit increase in x .
Examples
Exponential decay is a mathematical concept with numerous real-world applications. For instance, it accurately describes the depreciation of a car's value over time. Initially, the car's value is high, but as time passes, its value decreases, with each year seeing a smaller reduction in value than the previous year. This pattern aligns perfectly with exponential decay, where the rate of decrease slows down as the quantity diminishes. Understanding exponential decay helps in making informed decisions about when to sell or replace assets, optimizing financial planning and resource management.
