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!