Which statement holds true for absolute value functions? a) the absolute value determines the direction in which the graph opens. b) the coefficient determines the line along which the graph is symmetrical. c) the distance between the left and the right arm is based on the coordinates of the vertex. d) the vertex coordinates and the absolute value determine the region of the graph on the plane.

The correct answer is:
D) the vertex coordinates and the absolute value determine the region of the graph on the plane.

Explanation:

The vertex coordinates of an absolute value function determine how far the graph is shifted vertically and horizontally from the origin, as the vertex of the parent graph y=|x| is the origin (0, 0).

The coefficient in front of the absolute value bars determines which direction the graph opens up in as well as how wide or narrow the bars are (the distance between the arms).  If the coefficient is positive, the graph opens upward; if the coefficient is negative, the graph opens downward.  If the coefficient is an integer greater than 1 or less than -1, then the graph will be more narrow than the parent graph; if the coefficient is a fraction, then the graph will be wider than the parent graph.

Since the vertex affects the position and the coefficient determines the direction it opens up, these determine the region of the graph on the plane.