Question

compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply.
graph of a v - shaped absolute value function with vertex at (-2, -1), y - intercept around (0, 5), and passing through (-5, 6) and other points on a coordinate grid
\\(\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 Vertex Position

The parent function \( y = |x| \) has its vertex at \( (0,0) \). The given graph has its vertex at \( (-2, -1) \). Comparing the x - coordinates, the vertex has moved from \( x = 0 \) to \( x=- 2 \), which means there is a horizontal shift left (since moving from 0 to - 2 is a shift of 2 units to the left). Comparing the y - coordinates, the vertex has moved from \( y = 0 \) to \( y=-1 \), which means there is a vertical shift down.

Step2: Analyze Slope/Width

For the parent function \( y = |x| \), the slope of the right - hand side (for \( x\geq0 \)) is 1. Let's find the slope of the given graph. For the right - hand side of the given absolute - value graph, we can take two points. Let's take the vertex \( (-2,-1) \) and the point \( (0,5) \) (from the graph). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5-(-1)}{0 - (-2)}=\frac{6}{2}=3 \). Since the slope of the given graph (3) is greater than the slope of the parent function (1), the graph is narrower (a larger slope in the absolute - value function makes the graph narrower).

Step3: Analyze Reflection

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

Answer:

b. Horizontal Shift Left, e. Vertical Shift Down, f. Graph is Narrower