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

Explanation:

Step1: Recall parent - function properties

The parent - function of (h(x)) is (y = |x|), which has a vertex at ((0,0)) and opens upwards. The general form of a vertical translation of a function (y = f(x)) is (y=f(x)+k), where (k\gt0) shifts the graph of (y = f(x)) up by (k) units and (k\lt0) shifts it down by (|k|) units.

Step2: Analyze the given function

For the function (h(x)=|x| + 0.5), since (k = 0.5\gt0), the graph of (y = |x|) is shifted up by (0.5) units. The vertex of (y = |x|) is ((0,0)), and the vertex of (h(x)=|x|+0.5) is ((0,0.5)).

Answer:

The graph with the vertex at the point ((0,0.5)) and opening upwards.