The function you've provided is:

$h(x) = -2|x - 5|$

This is an absolute value function that has been vertically stretched and reflected across the x-axis. Let's break it down:

1. Absolute Value Function: The core part is $|x - 5|$, which means the function takes the absolute value of the expression inside the bars, ensuring that the output is always non-negative for this part.

2. Translation: The $(x - 5)$ indicates a horizontal shift of the function 5 units to the right. Normally, the vertex of the absolute value function $|x|$ is at the origin, but this shifts the vertex to $x = 5$.

3. Vertical Stretch and Reflection: The coefficient $-2$ outside the absolute value means two things:

   • Vertical Stretch: The factor of 2 stretches the graph vertically by a factor of 2.
   • Reflection: The negative sign reflects the graph across the x-axis. So instead of opening upwards (like a normal absolute value function), the graph will open downwards.

Key Points:

• The vertex of the function is at $(5, 0)$.
• The function decreases to the left of $x = 5$ and to the right, opening downward due to the negative coefficient.
• The slope of the lines on either side of the vertex is -2 and 2 (but since it's reflected, the lines will go down).

Graph Behavior:

• For $x < 5$, $h(x) = -2(5 - x)$, which is linear with a slope of 2 but reflected downwards.
• For $x \geq 5$, $h(x) = -2(x - 5)$, which is linear with a slope of -2.

Would you like more details or examples on how to work with this function?

Here are some related questions you might explore:

1. How do you find the vertex of an absolute value function?
2. What is the effect of changing the coefficient of the absolute value function?
3. How does shifting a function affect its graph?
4. Can you explain how to graph an absolute value function step by step?
5. What is the domain and range of this function?

Tip: The absolute value function is V-shaped, and the coefficient outside determines whether it opens up or down and how steeply it grows or shrinks.