Question

which of the following is the graph of this absolute value function?
y = |x + 2|

Explanation:

Step1: Recall the vertex form of absolute value function

The general form of an absolute value function is \( y = |x - h| + k \), where \((h, k)\) is the vertex of the V - shaped graph. For the function \( y=|x + 2|\), we can rewrite it as \( y=|x-(- 2)|+0\). So the vertex of the graph should be at \((h,k)=(-2,0)\).

Step2: Analyze the y - intercept

To find the y - intercept, we set \( x = 0\) in the function \( y=|x + 2|\). Then \( y=|0 + 2|=2\). So the graph should pass through the point \((0,2)\).

Step3: Analyze the three graphs

• For the first graph: The vertex is at \((-2,0)\) and it passes through \((0,2)\) (when \(x = 0\), \(y=|0 + 2| = 2\)). The left side of the V (for \(x<-2\)) has a negative slope and the right side (for \(x>-2\)) has a positive slope, which is consistent with the absolute value function \(y = |x+2|\).
• For the second graph: The vertex is at \((0,-2)\) and when \(x = 0\), \(y=|0 + 2|=2

eq - 2\), so this graph does not represent \(y = |x + 2|\).

• For the third graph: The vertex is at \((2,0)\) and when \(x=0\), \(y=|0 + 2| = 2\), but the vertex is at \(x = 2\) instead of \(x=-2\), so this graph does not represent \(y=|x + 2|\).

Answer:

The first graph (the left - most graph) is the graph of \(y = |x + 2|\)