Question

compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply.
graph of an absolute value function with vertex at (0, -2), passing through (-2, 0) and (2, 0), and points (-6, 4) and (6, 4)
options:
○ 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 vertex of the parent function and the given graph

The parent function \( y = |x| \) has its vertex at \( (0, 0) \). The given graph has its vertex at \( (0, -2) \).

Step2: Determine the vertical shift

A vertical shift down occurs when the vertex of the graph moves down along the y - axis. Since the vertex of the given graph is at \( (0, -2) \) compared to the parent function's vertex at \( (0, 0) \), the graph has a vertical shift down.

Step3: Analyze horizontal shift

The x - coordinate of the vertex of the parent function and the given graph is both 0, so there is no horizontal shift (neither left nor right).

Step4: Analyze reflection over x - axis

The parent function \( y = |x| \) opens upwards, and the given graph also opens upwards, so there is no reflection over the x - axis.

Step5: Analyze the width of the graph

The slope of the parent function \( y = |x| \) for \( x\geq0 \) is 1 (the equation is \( y = x \)) and for \( x < 0 \) is - 1 (the equation is \( y=-x \)). For the given graph, when \( x\geq0 \), let's take two points, say \( (2, 0) \) and \( (6, 4) \). The slope \( m=\frac{4 - 0}{6 - 2}=\frac{4}{4} = 1 \). When \( x<0 \), take two points \( (-6, 4) \) and \( (-2, 0) \), the slope \( m=\frac{0 - 4}{-2+6}=\frac{-4}{4}=-1 \). So the slope is the same as the parent function, which means the graph has the same width as the parent function (not narrower or wider).

Answer:

e. Vertical Shift Down