Question

without graphing, identify the vertex, axis of symmetry, and transformations from the parent function $f(x)=|x|$$\\ y=|x - 4| - 5$the vertex is $(4, -5)$(type an ordered pair.)the axis of symmetry is $x = 4$what are the transformations from the parent function?\\(\bigcirc\\) a. the function is translated 4 units to the right and vertically stretched by a factor of 5.\\(\bigcirc\\) b. the function is translated 4 units to the left and vertically stretched by a factor of 5.\\(\bigcirc\\) c. the function is translated 4 units down and vertically stretched by a factor of 5.\\(\bigcirc\\) d. the function is translated 4 units to the right and 5 units up.\\(\bigcirc\\) e. the function is translated 4 units to the left and 5 units down.\\(\bigcirc\\) f. the function is translated 4 units to the right and 5 units down.

Explanation:

Step1: Recall transformation rules

For absolute value function \( y = |x - h| + k \), horizontal shift: \( h \) units (right if \( h>0 \), left if \( h<0 \)), vertical shift: \( k \) units (up if \( k>0 \), down if \( k<0 \)). Stretching: coefficient of \( |x - h| \) (here coefficient is 1, no stretch).

Step2: Analyze \( y = |x - 4| - 5 \)

Compare with \( y = |x - h| + k \): \( h = 4 \), \( k = -5 \). So horizontal shift: 4 units right (since \( h = 4>0 \)), vertical shift: 5 units down (since \( k = -5<0 \)). No vertical stretch (coefficient 1).

Step3: Evaluate options

• A: No stretch (coefficient 1), wrong.
• B: \( h = 4 \) is right, not left, wrong.
• C: No stretch, wrong.
• D: \( k = -5 \) is down, not up, wrong.
• E: \( h = 4 \) is right, not left, wrong.
• F: 4 units right (\( h = 4 \)), 5 units down (\( k = -5 \)), correct.

Answer:

F. The function is translated 4 units to the right and 5 units down.