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The absolute value function is a fundamental mathematical function represented by the equation \( f(x) = |x| \). It is known for its distinct V-shaped graph, which is symmetric about the y-axis. This symmetry arises because the absolute value of a number is always non-negative, meaning any negative inputs are mirrored positively on the graph.

Here are some key characteristics of the absolute value function:

• The vertex, or the point where the two linear branches of the V meet, is at the origin, (0,0).
• The function increases at a rate of +1 on the right side of the vertex and decreases at a rate of -1 on the left.
• The graph of \( f(x) = |x| \) is always found in the first and second quadrants of the coordinate plane because the outputs are non-negative.

Understanding this foundational shape makes it easier to apply transformations, such as horizontal and vertical shifts, to modify its position on the graph.

A horizontal shift involves moving the graph of a function left or right on the coordinate plane. When considering transformations like those in \( h(x) = |x-1| + 4 \), horizontal shifts are affected by changes within the absolute value expression itself.

To determine the direction and magnitude of a horizontal shift, inspect the structure \( |x-a| \):

• If \( a \) is positive, the graph moves \( a \) units to the right.
• If \( a \) is negative, the graph moves \(|a| \) units to the left.

In our function, \( |x-1| \) signals a shift of 1 unit to the right. Therefore, the vertex of the graph moves from its original position at (0,0) to (1,0). A good way to confirm this shift is to substitute \( x = 0 \) and find the new location of the vertex. The process of shifting does not alter the V-shape structure, only its placement on the x-axis.

Vertical shifts occur when a constant is added or subtracted from the entire function, influencing its position along the y-axis. In the function \( h(x) = |x-1| + 4 \), the "+4" signifies a vertical shift upwards by 4 units.

Characteristics of vertical shifts include:

• An upward shift when the constant is positive.
• A downward shift when the constant is negative.

This shift results in adjusting the vertex from (1,0)—determined by the horizontal shift—to (1,4). It elevates the entire graph by 4 units without affecting the shape or orientation of the V. It's important to remember that vertical shifts do not influence the slope of the lines; they only adjust the position vertically.