Question

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

Explanation:

Step1: Recall the parent absolute value function

The parent absolute value function is \( y = |x| \), which has its vertex at \( (0, 0) \) and opens upwards (a V - shape).

Step2: Analyze the transformation

For the function \( y=|x| + 2 \), we are adding 2 to the parent function \( y = |x| \). In the transformation of functions, if we have a function \( y = f(x)+k \), when \( k>0 \), the graph of \( y = f(x) \) is shifted up by \( k \) units. Here, \( f(x)=|x| \) and \( k = 2 \), so the graph of \( y=|x| \) is shifted up by 2 units. This means the vertex of the graph of \( y=|x| + 2 \) will be at \( (0, 2) \).

Step3: Analyze the options

• The first graph has its vertex at \( (- 2,0) \), which is a horizontal shift, not a vertical shift of the parent function. So it is not the graph of \( y = |x|+2 \).
• The second graph has its vertex at \( (0, 2) \) and has the characteristic V - shape of an absolute value function, shifted up by 2 units from the parent function \( y = |x| \).
• The third graph has its vertex at \( (0,-2) \), which is a vertical shift down by 2 units, not up. So it is not the graph of \( y=|x| + 2 \).

Answer:

The second graph (the one with vertex at (0, 2) and V - shape opening upwards)