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

On a coordinate plane, an absolute value graph has a vertex at (0, 1.5).

On a coordinate plane, an absolute value graph has a vertex at (negative 0.5, 0).

On a coordinate plane, an absolute value graph has a vertex at (0, 0.5).

On a coordinate plane, an absolute value graph has a vertex at (negative 1.5, 0).

The function \( h(x) = |x| + 0.5 \) is an absolute value function. The graph of this function is a V-shaped graph with the vertex at the point where the expression inside the absolute value is zero. 

Given the function \( h(x) = |x| + 0.5 \):

- The vertex of this absolute value function is at \( (0, 0.5) \), because the graph of \( |x| \) is shifted vertically upwards by 0.5 units.

Therefore, the correct graph is:
- On a coordinate plane, an absolute value graph has a vertex at \( (0, 0.5) \).

Would you like more details or have any questions?

Here are some related questions to further your understanding:

1. What is the general form of an absolute value function?
2. How do vertical shifts affect the graph of an absolute value function?
3. What would be the vertex of the function \( h(x) = |x| - 2 \)?
4. How do you determine the vertex of the function \( h(x) = |x + 3| \)?
5. What is the effect of horizontal shifts on the graph of an absolute value function?
6. How does the graph of \( h(x) = -|x| \) differ from \( h(x) = |x| \)?
7. What are the characteristics of the graph of \( h(x) = |x| + c \) where \( c \) is a constant?
8. How can you find the axis of symmetry of an absolute value function?

**Tip:** The vertex of the function \( h(x) = |x - h| + k \) is at \( (h, k) \). Vertical shifts change the \( y \)-coordinate of the vertex, while horizontal shifts change the \( x \)-coordinate.

Understand how to graph the absolute value function h(x) = |x| + 0.5 with a focus on the vertex, vertical shifts, and graph characteristics. Suitable for high school students exploring functions and graph transformations.

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