Question

(a) which of the following is the correct graph of the function $y = \log_{2}(x + 3) - 1$?

Explanation:

Step1: Find Vertical Asymptote

For \( y = \log_2(x + 3)-1 \), the argument of log must be \(>0\), so \( x + 3>0\Rightarrow x>-3 \). The vertical asymptote is \( x=-3 \).

Step2: Find Key Point

Set \( x + 3 = 1\) (since \( \log_a(1)=0 \)), then \( x=-2 \). Substitute \( x = -2 \) into the function: \( y=\log_2(1)-1=0 - 1=-1 \). So the point \((-2, -1)\) is on the graph.

Step3: Analyze Graphs

• Check vertical asymptote \( x=-3 \): The first graph (A) has a vertical asymptote around \( x=-3 \) (since it approaches a vertical line near \( x=-3 \)), and the second graph (B) has a vertical asymptote left of \( x=-3 \) (incorrect).
• Check the point \((-2, -1)\): In graph A, at \( x=-2 \), \( y\approx -1 \), which matches. In graph B, the point at \( x=-2 \) is not \( -1 \). Also, the shape of \( y = \log_2(x + 3)-1 \) is a horizontal shift left 3 and vertical shift down 1 of \( y=\log_2x \), which matches graph A's shape (increasing, passing through \((-2, -1)\), asymptote \( x=-3 \)).

Answer:

A. The graph labeled A (the upper graph with vertical asymptote near \( x = -3 \) and passing through \((-2, -1)\))