Question

1. over what interval is the graph of $y = -|x - 2|$ increasing?

a. $(2, \infty)$
b. $(-\infty, 2)$
c. $(-\infty, -2)$
d. $(-2, \infty)$

Explanation:

Step1: Analyze the parent function

The parent function of absolute value is \( y = |x| \), which has a V - shape with vertex at \( (0,0) \), decreasing on \( (-\infty,0) \) and increasing on \( (0,\infty) \).

Step2: Analyze the transformation \( y=-|x - 2| \)

• The transformation \( y = |x - h| \) shifts the graph of \( y = |x| \) horizontally. For \( y=|x - 2| \), the vertex is at \( (2,0) \), and it is decreasing on \( (-\infty,2) \) and increasing on \( (2,\infty) \).
• The negative sign in front of the absolute value, \( y=-|x - 2| \), reflects the graph of \( y = |x - 2| \) over the x - axis. So the graph of \( y=-|x - 2| \) is a downward - opening V - shape with vertex at \( (2,0) \).
• For a downward - opening absolute value function \( y=-|x - 2| \), the function is increasing when the input \( x \) is in the interval where the inside of the absolute value function \( x - 2 \) is decreasing (because of the reflection). The function \( y = |x - 2| \) is decreasing on \( (-\infty,2) \), and after reflection, \( y=-|x - 2| \) will be increasing on \( (-\infty,2) \).

Answer:

B. \((-\infty,2)\)