Identify the equation that translates $y=\ln (x)$ five units down.
A. $y=\ln (x-5)$
B. $y=\ln (x)+5$
C. $y=\ln (x+5)$
D. $y=\ln (x)-5$
Answer (1)
To translate a function y = f ( x ) down by k units, we replace y with y + k , which gives us y = f ( x ) − k .
Substitute f ( x ) = ( x ) and k = 5 into the equation.
The translated function is y = ( x ) − 5 .
The equation that translates y = ( x ) five units down is y = ( x ) − 5 .
Explanation
Understanding the Problem The problem asks us to find the equation of the function y = ( x ) after it has been translated 5 units down.
Vertical Translation To shift a function y = f ( x ) vertically downwards by k units, we subtract k from the function's value. In other words, the new function becomes y = f ( x ) − k .
Applying the Translation In our case, f ( x ) = ( x ) and k = 5 . Therefore, the translated function is y = ( x ) − 5 .
Final Answer The equation that translates y = ( x ) five units down is y = ( x ) − 5 .
Examples
Imagine you're tracking the height of a plant over time, and you notice that the plant's growth is consistently offset by a certain amount due to environmental factors. Translating the growth curve downwards allows you to isolate the plant's intrinsic growth pattern from these external influences, providing a clearer picture of its biological development. This concept applies to various fields, such as adjusting stock prices for inflation or calibrating sensor readings in engineering.
