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)