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An absolute value function, represented as \(|x|\), provides the non-negative value of \(x\).
In simple terms, whether our \(x\) value is positive or negative, the result is always positive. This concept visually translates to a V-shape graph in a 2-dimensional space.
Let's dive deeper into how this looks. For a function like \(f(x) = |x-1|\), the absolute value part \(|x-1|\) moves the vertex of the V-shaped graph to the point \(x=1\), creating two distinct linear parts. Each piece of the function is a line:

• One line extends infinitely towards the positive x-direction.
• The other extends towards the negative x-direction.

In this exercise, the function is tasked with the negative of an absolute value: \(f(x) = -|x-1|\). This flips the V-shape upside down. Having a negative sign makes each point of the graph mirrored over the x-axis, resulting in an inverted V-shape. Each part of the function still forms a line, making it easier to pinpoint the vertex and calculate the slopes of each segment.

Graph analysis involves studying a graph to understand the behavior and properties of the function it represents.
In the case of \(f(x) = -|x-1|\), analyzing the graph helps in determining where the function has specific characteristics like direction change or slope.The vertex of our graph, \(x = 1\), acts as the pivotal point where the function's direction changes. This is where the graph goes from having a negative slope to a positive slope.
Graphically, the inverted V created by \(f(x) = -|x-1|\) demonstrates that:

• The function slopes downward from the left until it reaches the vertex at \(x = 1\).
• Beyond \(x = 1\), the graph slopes upwards.

This information is critical to understand because it affects where the derivative of the function, \(f'(x)\), will exist or be undefined. The different slope sections indicate different rates of change or derivative, and at the vertex, the derivative doesn't exist due to the abrupt change in direction.

The slope of a line is essentially the measure of its steepness. When dealing with functions like \(f(x) = -|x-1|\), each linear segment of the graph has its constant slope except at points like vertices.For our given function:

• To the left of the vertex \((x < 1)\), the slope is calculated from the line \(f(x) = -(x-1) = -x+1\), resulting in a slope \(f'(x) = -1\). This negative value indicates the line is slanting downwards.
• To the right of the vertex \((x > 1)\), the slope comes from the line \(f(x) = x-1\), giving us \(f'(x) = 1\). Here, the positive slope signifies the line is slanting upwards.

Therefore, understanding how these slopes work assists in visualizing how the function behaves and where the derivative \(f'(x)\) holds value. At the vertex, \(x=1\), the slope transitions sharply—making it undefined because you can't suddenly leap from a negative to a positive slope without crossing smooth terrain. That's why the derivative does not exist at this critical point in the graph.