Question

1. use the given table of values to help you select the graph of the absolute value function $f(x) = -|x - 1|$

$x$	$y$
$0$	$-1$
$1$	$0$
$2$	$-1$
$3$	$-2$

(graph with coordinate grid and a v - shaped line)

Explanation:

Step1: Analyze the function's vertex

The absolute value function \( f(x) = -|x - 1| \) has its vertex at \( (h, k) \) where for \( f(x)=a|x - h|+k \), here \( h = 1 \), \( k = 0 \), and \( a=- 1 \) (which means it opens downward). The table has a point \( (1,0) \), so the vertex should be at \( (1,0) \).

Step2: Check the graph's vertex and direction

The given graph in the image has its vertex at \( (0,0) \) (since it touches the origin) and opens upward (both arms go up). But our function \( f(x)=-|x - 1| \) should have vertex at \( (1,0) \) and open downward (because \( a=-1<0 \)). Wait, but maybe there are other graphs (not shown here? Wait, the user's image has one graph, but maybe the correct graph should have vertex at \( (1,0) \), opens downward, and passes through the points from the table: \( (-1,-2) \), \( (0,-1) \), \( (1,0) \), \( (2,-1) \), \( (3,-2) \). Let's check the table points: when \( x = 1 \), \( y = 0 \) (vertex). For \( x = 0 \), \( y=-1 \); \( x = 2 \), \( y=-1 \); \( x=-1 \), \( y = - 2 \); \( x = 3 \), \( y=-2 \). So the graph should be a V - shaped graph opening downward with vertex at \( (1,0) \), passing through those points. The shown graph has vertex at \( (0,0) \) and opens upward, so it's not correct. But since the problem is to select the graph, maybe there are other options, but based on the table, the correct graph should have vertex at \( (1,0) \), open downward, and pass through the given \( (x,y) \) points.

Wait, maybe the user's image has only one graph, but perhaps the correct graph (not shown here) would have vertex at \( (1,0) \), and the arms going down. But since the current graph has vertex at \( (0,0) \) and opens up, it's incorrect. But maybe the question is to identify that the given graph is not correct, or maybe there are other graphs. Wait, the problem says "use the given table of values to help you select the graph". Let's re - evaluate:

The function \( f(x)=-|x - 1| \) is a transformation of \( y = |x| \). The parent function \( y = |x| \) has vertex at \( (0,0) \), opens up. \( y=-|x| \) opens down, vertex at \( (0,0) \). \( y=-|x - 1| \) shifts \( y=-|x| \) 1 unit to the right, so vertex at \( (1,0) \), opens down.

The table has \( (1,0) \) (vertex), \( (0,-1) \) (since \( f(0)=-|0 - 1|=-1 \)), \( (2,-1) \) ( \( f(2)=-|2 - 1|=-1 \) ), \( (-1,-2) \) ( \( f(-1)=-|-1 - 1|=-2 \) ), \( (3,-2) \) ( \( f(3)=-|3 - 1|=-2 \) ). So the graph should have vertex at \( (1,0) \), and for \( x < 1 \), as \( x \) increases towards 1, \( y \) increases towards 0 (since when \( x \) goes from - 1 to 1, \( x \) increases, \( |x - 1| \) decreases, so \( -|x - 1| \) increases). For \( x>1 \), as \( x \) increases, \( |x - 1| \) increases, so \( -|x - 1| \) decreases. So the graph is a downward - opening V with vertex at \( (1,0) \), passing through \( (0,-1) \), \( (2,-1) \), \( (-1,-2) \), \( (3,-2) \).

The given graph in the image has vertex at \( (0,0) \), opens up, so it's not the graph of \( f(x)=-|x - 1| \). But since the problem is to select the graph, and maybe the correct graph (not shown here) would have vertex at \( (1,0) \), open downward, and pass through the table's points.

But maybe the user made a mistake in the image, but based on the table, the correct graph should have vertex at \( (1,0) \), and the shape as per the function's transformation.

Answer:

The graph with vertex at \( (1,0) \), opening downward, and passing through the points \((-1, - 2)\), \((0, - 1)\), \((1,0)\), \((2, - 1)\), \((3, - 2)\) (not the shown graph with vertex at \((0,0)\) opening upward). If we assume there are multiple graphs, the correct one will have vertex at \( (1,0) \), open downward, and pass through the table's points.