The function you've provided is:

\[
f(x) = -|x - 5| + 4
\]

This is a transformation of the absolute value function \( f(x) = |x| \). Let's break down the transformations step by step:

### 1. Absolute Value Function
The base function is \( |x - 5| \), which has a "V" shape. The vertex of the "V" occurs where \( x = 5 \), since the absolute value is zero at \( x = 5 \).

### 2. Reflection
The negative sign in front of the absolute value function, \( -|x - 5| \), reflects the graph over the x-axis. Instead of opening upwards like a typical absolute value graph, it opens downwards.

### 3. Vertical Shift
The \( +4 \) at the end shifts the entire graph upward by 4 units. This moves the vertex from \( (5, 0) \) to \( (5, 4) \).

### Key Characteristics:
- **Vertex:** \( (5, 4) \)
- **Opening:** Downwards (due to the negative sign)
- **Domain:** All real numbers, \( (-\infty, \infty) \)
- **Range:** The function takes values less than or equal to 4, so the range is \( (-\infty, 4] \).

Would you like me to graph this function or go into more detail about any part of the transformation?

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Here are five related questions you might want to explore:
1. How would the graph change if the negative sign was removed?
2. What is the effect of shifting the vertex to a different point?
3. How do you find the domain and range of other absolute value functions?
4. Can you solve \( f(x) = 0 \) for this particular function?
5. How can this type of function be applied in real-world scenarios?

**Tip:** To graph absolute value functions, always start by locating the vertex, as it helps set the shape and direction of the graph!

Explore the graph and transformations of the function f(x) = -|x - 5| + 4. This step-by-step breakdown covers absolute value functions, reflections, and vertical shifts. Suitable for students in Grades 9-11.

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