Question

the graph of a logarithmic function is shown. select the function which matches the graph. a. $y = \log_3 x + 2$ b. $y = \log_3 x$ c. $y = \log_3 (x + 2)$ d. $y = \log_3 (x - 2)$

Explanation:

Step1: Recall Logarithmic Shift Rules

For a logarithmic function \( y = \log_b(x - h) + k \), horizontal shift is determined by \( h \): shift left if \( h < 0 \) (replace \( h \) with \( -|h| \), so \( x + |h| \)), shift right if \( h > 0 \). Vertical shift is \( k \). The parent function \( y = \log_3 x \) has vertical asymptote \( x = 0 \).

Step2: Analyze the Graph's Asymptote

The graph's vertical asymptote is at \( x = -2 \) (from the grid, the curve approaches \( x = -2 \)). For \( y = \log_3(x + 2) \), set \( x + 2 = 0 \Rightarrow x = -2 \), which matches the asymptote.

• Option A: \( y = \log_3 x + 2 \) has asymptote \( x = 0 \), wrong.
• Option B: \( y = \log_3 x \) has asymptote \( x = 0 \), wrong.
• Option C: \( y = \log_3(x + 2) \) has asymptote \( x = -2 \), correct.
• Option D: \( y = \log_3(x - 2) \) has asymptote \( x = 2 \), wrong.

Answer:

C. \( y = \log_3(x + 2) \)