Question

select the correct answer.
the graph of function f is shown.
which statement correctly describes the graph of g(x) = f(x - 9)?
a. function g has the same horizontal and vertical asymptotes as function f.
b. function g has the same horizontal asymptote as f and a vertical asymptote at x = 6.
c. function g has the same horizontal asymptote as f and a horizontal asymptote at x = 9.
d. function g has the same vertical asymptote as f and a horizontal asymptote at y = -5.

Explanation:

Step1: Analyze Vertical Asymptote of f

From the graph, the vertical asymptote of \( f(x) \) is at \( x = -3 \) (assuming the dashed vertical line is at \( x=-3 \)). For \( g(x)=f(x - 9) \), the transformation \( x\to x - 9 \) shifts the graph horizontally. The vertical asymptote of \( g(x) \) is found by solving \( x - 9=-3 \), so \( x = 6 \).

Step2: Analyze Horizontal Asymptote of f and g

The horizontal asymptote of \( f(x) \) (from the graph, the dashed horizontal line) is \( y = 3 \) (or similar, but the key is horizontal asymptote is a \( y \)-value). Horizontal asymptotes are not affected by horizontal shifts, so \( g(x) \) has the same horizontal asymptote as \( f(x) \).

Step3: Evaluate Options

• Option A: Vertical asymptote changes, so A is wrong.
• Option B: Horizontal asymptote same, vertical asymptote at \( x = 6 \) (from \( x-9=-3\Rightarrow x = 6 \)), so B is correct.
• Option C: Vertical asymptote is not horizontal, so C is wrong.
• Option D: Horizontal asymptote is \( y \)-value, not \( x = 9 \), and vertical asymptote changes, so D is wrong.

Answer:

B. Function g has the same horizontal asymptote as f and a vertical asymptote at \( x = 6 \).