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: Recall the transformation rule

The transformation \(y = f(x - h)\) is a horizontal shift of the graph of \(y = f(x)\) to the right by \(h\) units.

Step2: Analyze the asymptotes

For a function \(y = f(x)\) and its transformation \(y=f(x - 9)\) (where \(h = 9\)):

• The horizontal asymptote of a function \(y = f(x)\) is determined by the end - behavior of the function. Since \(g(x)=f(x - 9)\) is a horizontal shift, the end - behavior (which determines the horizontal asymptote) does not change.
• The vertical asymptote of \(y = f(x)\) occurs at \(x=a\). For \(y = f(x - 9)\), we set \(x-9=a\). If the vertical asymptote of \(y = f(x)\) is at \(x=- 3\) (from the graph, assume the vertical asymptote of \(y = f(x)\) is at \(x=-3\)), then for \(y = f(x - 9)\), we solve \(x-9=-3\) gives \(x = 6\)

Answer:

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