the function $y = \\log(x)$ is translated 1 unit right and 2 units down. which is the graph of the…

the function $y = \\log(x)$ is translated 1 unit right and 2 units down. which is the graph of the translated function?

the function $y = \\log(x)$ is translated 1 unit right and 2 units down. which is the graph of the translated function?

Answer

Explanation:

Step1: Recall transformation rules

For a function ( y = f(x) ), shifting ( h ) units right and ( k ) units down gives ( y = f(x - h) - k ). Here, ( f(x)=\log(x) ), ( h = 1 ), ( k = 2 ), so the translated function is ( y=\log(x - 1)-2 ).

Step2: Analyze key points

  • Vertical asymptote: For ( y=\log(x) ), asymptote is ( x = 0 ). After translation, asymptote becomes ( x=1 ) (since ( x - 1=0\Rightarrow x = 1 )).
  • Intercepts: When ( x = 2 ), ( y=\log(2 - 1)-2=\log(1)-2=0 - 2=-2 ). So the graph passes through ( (2, - 2) ).
  • Behavior: The original ( \log(x) ) increases slowly. After translation, the shape remains logarithmic, shifted right 1 and down 2. The given graph (the one with the curve starting near ( x = 1 ), passing through ( (2, - 2) ) and increasing) matches this.

Answer:

The graph shown (the one with the curve starting near ( x = 1 ), passing through ( (2, - 2) ) and increasing as ( x ) increases) is the graph of ( y=\log(x - 1)-2 ). (If we consider the provided graph with the curve starting at ( x\approx1 ), going through ( (2, - 2) ), that's the correct one.)