Question

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: Recall Transformations

For a function \( y = f(x - h) \), it's a horizontal shift of \( f(x) \) by \( h \) units (right if \( h>0 \), left if \( h<0 \)). Horizontal shifts don't affect horizontal asymptotes (HA) but shift vertical asymptotes (VA) by \( h \) units.

Step2: Analyze Original Graph

From the graph of \( f(x) \), identify VA and HA. The VA of \( f(x) \) is at \( x = -3 \) (since the left branch approaches a vertical line, likely \( x=-3 \)), and HA: observe the horizontal behavior, say \( y = 2 \) (or whatever the HA is, but key is horizontal shift doesn't change HA).

Step3: Apply Shift to VA

For \( g(x) = f(x - 9) \), shift VA of \( f(x) \) ( \( x=-3 \)) right by 9 units: \( -3 + 9 = 6 \). HA remains same as \( f(x) \) (horizontal shifts don't affect HA).

Step4: Evaluate Options

• Option A: VA changes, so wrong.
• Option B: HA same, VA at \( x=6 \) (shifted from \( x=-3 \) by +9), correct.
• Option C: VA is vertical, not horizontal, wrong.
• Option D: HA and VA descriptions wrong.

Answer:

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