Question

compare the graph to the absolute value parent function (y = |x|). select all the transformations that apply

a reflected over the x - axis
b horizontal shift left
c horizontal shift right
d vertical shift up
e vertical shift down
f graph is narrower
g graph is wider

Explanation:

Step1: Analyze the parent function \( y = |x| \)

The parent function \( y = |x| \) has a vertex at \( (0,0) \), opens upwards, and has a slope of \( 1 \) for \( x \geq 0 \) and \( -1 \) for \( x < 0 \).

Step2: Analyze the given graph

• Vertex: The vertex of the given graph is at \( (0, 3) \), while the parent function's vertex is at \( (0,0) \). This indicates a vertical shift up by \( 3 \) units (so option d applies).
• Direction: The parent function opens upwards, and the given graph also opens upwards, so no reflection over the x - axis (option a does not apply).
• Horizontal Shift: The vertex is still on the y - axis (\( x = 0 \)), so there is no horizontal shift (options b and c do not apply).
• Width/Steepness: The slope of the given graph for \( x\geq0 \): Let's take two points, say \( (0,3) \) and \( (3,0) \). The slope \( m=\frac{0 - 3}{3-0}=- 1 \), same as the parent function \( y = |x| \) (slope \( 1 \) for \( x\geq0 \) and \( - 1 \) for \( x < 0 \)). So the graph has the same width (not narrower or wider, so options f and g do not apply). Also, since the vertex is at \( (0,3) \) instead of \( (0,0) \), it is a vertical shift up.

Answer:

d. Vertical Shift Up