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The absolute value function is a fundamental concept in mathematics that describes a function's distance from zero. The absolute value of a number is always non-negative. Specifically, the absolute value function transforms any input into its non-negative equivalent. For a given function, such as our exercise's function, \( y = |x-2| \), the absolute value affects how the graph behaves.

The absolute value function creates a V-shaped graph:

• It appears linear but forms two straight lines with a vertex, or a sharp corner.
• At the vertex, the graph changes direction.
• It is symmetric on either side of the line reaching the vertex.

For the function \( y = |x-2| \), the value of \( x \) that makes the expression \( x-2 \) zero is \( x = 2 \). Here, the function's graph has a vertex at this point. This critical moment defines where the absolute value impacts the function the most, as it shifts the direction of the line.

Graphing functions visually represents mathematical equations and provides a clear picture of their behavior. For absolute value functions like \( y = |x-2| \), the graph shows a V-shape. This is due to the nature of absolute values, which reflect any negative input across the x-axis into positive outputs. Here's a brief overview of the graphing steps involved:

For \( y = |x-2| \):

• Calculate how the graph behaves on either side of the vertex. On either side of \( x=2 \), the function switches from \( y = -(x-2) = -x+2 \) to \( y = (x-2) = x-2 \).
• Plot the vertex as a key point, knowing it's where the direction changes.
• Draw the linear paths on either side of the vertex.

By analyzing and plotting in such a way:

• The visual aspect quickly reveals the shape and behavior of the function.
• It shows crucial information like continuity and differentiability points.

Graphing is vital in understanding where a function changes direction (like in absolute values) and predicting precise function behavior.

Solving problems in calculus involves understanding key principles like continuity and differentiability, especially for functions like \( y = |x-2| \). Differentiability is an important aspect to consider. It refers to a function's ability to have a defined derivative at a certain point:

• For \( y = |x-2| \), the graph is smooth on both sides but has a corner at \( x=2 \).
• This corner means that the derivative (slope) changes abruptly.
• Thus, it is non-differentiable at this specific point.

Additionally,

When solving calculus problems, we look for:

• Points where the derivative may not exist due to sharp points or discontinuities.
• Critical points, such as \( x = 2 \) in our example, where the function might not align smoothly.

Understanding these principles helps in graphing functions, predicting outcomes, and solving complex calculus problems with greater confidence.