Question

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

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.)