Question

question 4 (1 point)
compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply.
graph of a coordinate plane with a blue absolute - value - shaped graph
$\square$ a reflected over the x - axis
$\square$ b horizontal shift left
$\square$ c horizontal shift right
$\square$ d vertical shift up
$\square$ e vertical shift down
$\square$ f graph is narrower
$\square$ g graph is wider

Explanation:

Step1: Analyze Reflection

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

Step2: Analyze Horizontal Shift

The vertex of the parent function \( y = |x| \) is at \( (0,0) \). The vertex of the given graph is at \( (-2,-6) \). For horizontal shift, if the vertex moves from \( x = 0 \) to \( x=-2 \), this is a horizontal shift left by 2 units. So option b is correct and option c is incorrect.

Step3: Analyze Vertical Shift

The y - coordinate of the vertex of the parent function is \( y = 0 \), and for the given graph, the y - coordinate of the vertex is \( y=-6 \). So the graph has a vertical shift down by 6 units. So option e is correct and option d is incorrect.

Step4: Analyze Width (Stretch/Compression)

The slope of the parent function \( y = |x| \) (for \( x\geq0 \)) is 1. For the given graph, let's take two points. For example, when \( x = 0 \), \( y=-5\) (approximate from the graph) and when \( x = 2 \), \( y=-3\). The slope between \( (0,-5) \) and \( (2,-3) \) is \( \frac{-3 + 5}{2-0}=\frac{2}{2}=1 \), same as the parent function. So there is no horizontal or vertical stretch/compression, so options f and g are incorrect.

Answer:

b. Horizontal Shift Left, e. Vertical Shift Down