Question

which graph represents the function $h(x) = |x| + 0.5$?

Explanation:

Step1: Analyze the parent function

The parent function is \( y = |x| \), which has a vertex at \( (0, 0) \) and is a V - shaped graph opening upwards.

Step2: Analyze the transformation

The function \( h(x)=|x| + 0.5 \) is a vertical shift of the parent function \( y = |x| \). For a function of the form \( y=f(x)+k \), if \( k>0 \), the graph of \( y = f(x) \) is shifted up by \( k \) units. Here, \( k = 0.5>0 \), so the vertex of the graph of \( h(x) \) should be at \( (0,0 + 0.5)=(0,0.5) \). We also know that the slope of the right - hand side (for \( x\geq0 \)) of the absolute - value function \( y = |x| \) is 1, and the slope of the left - hand side (for \( x<0 \)) is - 1. So we need to find the graph with a vertex at \( (0,0.5) \) and the correct slope on both sides.
Looking at the graphs:

• The first graph has a vertex at \( (0,1.5) \) (incorrect vertical shift).
• The second graph has a vertex at \( (0,0) \) (no vertical shift, incorrect).
• The third graph has a vertex at \( (0,0.5) \) and the correct V - shape with slopes 1 (for \( x\geq0 \)) and - 1 (for \( x<0 \)).
• The fourth graph has a vertex at \( (- 1.5,0) \) (incorrect horizontal and vertical shift).

Answer:

The third graph (the one in the top - right among the first three and above the fourth graph)