Identify the equation that translates [tex]y=\ln (x)[/tex] five units down. [tex]y=\ln (x-5)[/tex] [tex]y=\ln (x)+5[/tex] [tex]y=\ln (x+5)[/tex] [tex]y=\ln (x)-5[/tex]
Answer (2)
To shift a function's graph down by k units, subtract k from the function's expression.
Given the original equation y = ( x ) , we want to shift it down by 5 units.
The translated equation is y = ( x ) − 5 .
Therefore, the equation that translates y = ( x ) five units down is y = ( x ) − 5 .
Explanation
Understanding the Problem The problem asks us to identify the equation that represents the graph of y = ( x ) shifted five units down.
Vertical Shifts To shift a function's graph vertically, we add or subtract a constant from the function's expression. A downward shift corresponds to subtracting a constant.
Applying the Shift If we want to shift the graph of y = f ( x ) down by k units, the new equation is y = f ( x ) − k . In our case, f ( x ) = ( x ) and k = 5 . Therefore, the translated equation is y = ( x ) − 5 .
Final Answer The equation that translates y = ( x ) five units down is y = ( x ) − 5 .
Examples
Imagine you are tracking the population growth of a bacteria colony using a logarithmic scale, y = ( x ) , where x is time. If a sudden environmental change causes the initial population to decrease, you might want to adjust your model to reflect this downward shift. Subtracting a constant, such as 5, from the original equation gives you y = ( x ) − 5 , which represents the new population model after the environmental change. This adjustment helps you accurately predict future population sizes based on the altered starting point.
The equation that translates y = ln ( x ) five units down is y = ln ( x ) − 5 . This is achieved by subtracting 5 from the original function. The correct choice from the options provided is y = ln ( x ) − 5 .
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