Question

what does the transformation $f(x) \mapsto f(x - 10) - 4$ do to the graph of $f(x)$? translates it 10 units left and 4 units down translates it 10 units right and 4 units up translates it 10 units right and 4 units down translates it 10 units left and 4 units up

Explanation:

Step1: Recall horizontal translation rule

For a function \( y = f(x) \), the transformation \( f(x - h) \) represents a horizontal translation. If \( h>0 \), it is a translation \( h \) units to the right; if \( h < 0 \), it is a translation \( |h| \) units to the left. Here, we have \( f(x-10) \), so \( h = 10>0 \), which means a translation 10 units to the right.

Step2: Recall vertical translation rule

For a function \( y = f(x) \), the transformation \( f(x)+k \) represents a vertical translation. If \( k>0 \), it is a translation \( k \) units up; if \( k < 0 \), it is a translation \( |k| \) units down. Here, we have \( f(x - 10)-4=f(x - 10)+(- 4) \), so \( k=-4<0 \), which means a translation 4 units down.

Answer:

translates it 10 units right and 4 units down