Question

without graphing, identify the vertex, axis of symmetry, and transformations from the parent function f(x) = |x|.

y = |x + 1| + 2

the vertex is (-1,2).
(type an ordered pair.)

the axis of symmetry is x = -1.

what are the transformations from the parent function?

a. the function is translated 1 unit up and vertically stretched by a factor of 2.

b. the function is translated 1 unit to the right and vertically stretched by a factor of 2.

c. the function is translated 2 units to the left and 1 unit up.

d. the function is translated 1 unit to the left and 2 units up.

e. the function is translated 1 unit to the left and vertically stretched by a factor of 2.

f. the function is translated 2 units to the right and 1 unit down.

Explanation:

Step1: Recall Transformation Rules

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

Step2: Analyze \( y = |x + 1| + 2 \)

Rewrite as \( y = |x - (-1)| + 2 \). So \( h = -1 \) (left 1 unit), \( k = 2 \) (up 2 units). No vertical stretch (coefficient 1).

Step3: Evaluate Options

• A: Wrong (no stretch, translation up 2, left 1).
• B: Wrong (left 1, not right, no stretch).
• C: Wrong (left 1, up 2, not left 2, up 1).
• D: Correct (left 1, up 2).
• E: Wrong (no stretch).
• F: Wrong (directions and amounts wrong).

Answer:

D. The function is translated 1 unit to the left and 2 units up.