Question

what does the transformation $f(x) \mapsto f(x + 6) + 9$ do to the graph of $f(x)$? translates it 6 units left and 9 units up translates it 6 units right and 9 units up translates it 6 units left and 9 units down translates it 6 units right and 9 units down

Explanation:

To determine the transformation of the graph of \( f(x) \) under \( f(x) \to f(x + 6)+9 \), we use the rules of function transformations:

• For horizontal shifts: The transformation \( f(x)\to f(x + h) \) shifts the graph \( h \) units to the left if \( h>0 \) and \( h \) units to the right if \( h < 0 \). Here, \( h = 6>0 \), so the graph is shifted 6 units to the left.
• For vertical shifts: The transformation \( f(x)\to f(x)+k \) shifts the graph \( k \) units up if \( k>0 \) and \( k \) units down if \( k < 0 \). Here, \( k=9 > 0 \), so the graph is shifted 9 units up.

Combining these two transformations, the graph of \( f(x) \) is translated 6 units left and 9 units up.

Answer:

A. translates it 6 units left and 9 units up