The function $f(x)=\left(\frac{1}{5}\right)^x$ is translated up 4 units. Which equation represents the translated function?
A. $g(x)=\left(\frac{1}{5}\right)^{x-4}$
B. $g(x)=\left(\frac{1}{5}\right)^{x+4}$
C. $g(x)=\left(\frac{1}{5}\right)^x+4$
D. $g(x)=\left(\frac{1}{5}\right)^x-4$
Answer (2)
To translate a function f ( x ) up by k units, we add k to the function: g ( x ) = f ( x ) + k .
Given f ( x ) = ( 5 1 ) x and a translation of 4 units upwards, we have k = 4 .
Therefore, the translated function is g ( x ) = ( 5 1 ) x + 4 .
The equation representing the translated function is g ( x ) = ( 5 1 ) x + 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( 5 1 ) x and asked to find the equation of the function g ( x ) that results from translating f ( x ) up by 4 units.
Vertical Translation To translate a function f ( x ) vertically by k units, we add k to the function. In other words, the translated function g ( x ) is given by g ( x ) = f ( x ) + k .
Applying the Translation In this case, we want to translate f ( x ) = ( 5 1 ) x up by 4 units, so k = 4 . Therefore, the translated function is g ( x ) = f ( x ) + 4 = ( 5 1 ) x + 4.
Final Answer The equation that represents the translated function is g ( x ) = ( 5 1 ) x + 4 .
Examples
Imagine you're tracking the decay of a radioactive substance, modeled by an exponential function. If you want to account for a constant background radiation level, you'd translate the decay function upwards. Similarly, in economics, if you have a cost function and want to account for a fixed additional cost (like rent), you'd translate the cost function upwards. Understanding function translations helps model real-world scenarios with added constants.
The translated function of f ( x ) = ( 5 1 ) x when moved up by 4 units is g ( x ) = ( 5 1 ) x + 4 , which corresponds to option C. This means we add 4 to the original function. Therefore, the answer is g ( x ) = ( 5 1 ) x + 4 .
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