Question

question 9 (1 point)
compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply.
graph of a v - shaped function with vertex at (0,0), passing through (-2,-2) and (2,-2)
□ 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 \( (0,0) \), and for \( x = 1 \), \( y = 1 \); for \( x = -1 \), \( y = 1 \). Its graph is a V - shape opening upwards with a slope of \( 1 \) for \( x>0 \) and \( - 1 \) for \( x < 0 \).

Step2: Analyze the given graph

• Reflection over x - axis: The parent function \( y = |x| \) opens upwards. The given graph opens downwards. A reflection over the \( x \) - axis of a function \( y = f(x) \) gives \( y=-f(x) \). For \( y = |x| \), reflecting over the \( x \) - axis gives \( y=-|x| \), which matches the direction of the given graph (opening downwards).
• Vertical shift: The vertex of the parent function \( y = |x| \) is at \( (0,0) \). The vertex of the given graph is also at \( (0,0) \), so there is no vertical shift (up or down).
• Horizontal shift: The vertex of the given graph is at \( (0,0) \), same as the parent function \( y = |x| \), so there is no horizontal shift (left or right).
• Width of the graph: For the parent function \( y = |x| \), the slope of the right - hand side (for \( x>0 \)) is \( 1 \). Let's check the slope of the given graph. For \( x = 2 \), the \( y \) - value is \( - 2 \) (from the graph). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}\), taking two points on the right - hand side, say \( (0,0) \) and \( (2,-2) \), the slope \( m=\frac{-2 - 0}{2 - 0}=- 1 \). The absolute value of the slope is \( 1 \), same as the parent function. So the graph is not narrower or wider.

Answer:

a. Reflected over the x - axis