Question

question
the graph of ( y = f(x) ) is the solid black graph below. which function represents the dotted graph?
answer
( circ y = f(x + 3) - 1 )
( circ y = f(x + 3) + 1 )
( circ y = f(x - 3) - 1 )
( circ y = f(x - 3) + 1 )

Explanation:

Step1: Analyze Horizontal Shift

To determine the horizontal shift, we look at the direction the graph moves. If the dotted graph is shifted to the right by 3 units from the solid graph \( y = f(x) \), the transformation for horizontal shift is \( f(x - h) \) where \( h \) is the shift. A shift right by 3 means \( h = 3 \), so the horizontal part is \( f(x - 3) \).

Step2: Analyze Vertical Shift

For the vertical shift, if the dotted graph is shifted down by 1 unit from the solid graph, the transformation for vertical shift is \( f(x) + k \) where \( k \) is the shift. A shift down by 1 means \( k=-1 \), so the vertical part is \( - 1 \).

Combining horizontal (right 3) and vertical (down 1) shifts, the function is \( y = f(x - 3)-1 \).

Answer:

\( y = f(x - 3)-1 \) (the option with this formula, e.g., if the option is labeled with this, that's the answer)