To describe the translation of the function [tex]\( f(x) = |x| \)[/tex] to [tex]\( f(x) = |x+5| - 3 \)[/tex], let's carefully analyze the transformations step by step.
1. Horizontal Shift:
- The expression inside the absolute value has changed from [tex]\( x \)[/tex] to [tex]\( x + 5 \)[/tex].
- Generally, [tex]\( f(x + c) \)[/tex] translates the graph of [tex]\( f(x) \)[/tex] horizontally by [tex]\( c \)[/tex] units:
- If [tex]\( c \)[/tex] is positive, the shift is to the left.
- If [tex]\( c \)[/tex] is negative, the shift is to the right.
- In this case, [tex]\( c = 5 \)[/tex], so it indicates a horizontal shift [tex]\( 5 \)[/tex] units to the left.
2. Vertical Shift:
- The entire expression outside the absolute value has a constant [tex]\( -3 \)[/tex] added to it.
- Generally, [tex]\( f(x) + k \)[/tex] translates the graph of [tex]\( f(x) \)[/tex] vertically by [tex]\( k \)[/tex] units:
- If [tex]\( k \)[/tex] is positive, the shift is upward.
- If [tex]\( k \)[/tex] is negative, the shift is downward.
- In this case, [tex]\( k = -3 \)[/tex], so it indicates a vertical shift [tex]\( 3 \)[/tex] units downward.
Combining both shifts together:
- The function [tex]\( y = |x+5| - 3 \)[/tex] indicates a horizontal shift 5 units to the left and a vertical shift 3 units downward.
Therefore, the correct translation of the function is:
- 5 units left, 3 units down
So, the answer is:
[tex]\[ \boxed{\text{5 units left, 3 units down}} \][/tex]
