91Ó°ÊÓ

The absolute value function, like the one expressed as \(y=|x|\), is a common and foundational type of mathematical graph. It is characterized by its distinctive V-shaped structure. This V-shape occurs because the absolute value function takes any negative input and reflects it across the x-axis into a positive output. Mathematically, this means for any non-negative x-values, \(y = x\); and for negative x-values, \(y = -x\). Thus, it results in a graph where the vertex or lowest point is at the origin, for the function \(y=|x|\). To quickly sketch this function, you can plot key points such as \((-1,1)\), \((0,0)\), and \((1,1)\) and draw straight lines connecting them, forming the characteristic V shape.

Piecewise functions are composed of different expressions, depending on the input value (x) range. A classic example is the function \(y = \frac{1}{2}(x + |x|)\). Here, the function behaves differently for positive and negative x-values. Specifically, when \(x \geq 0\), you have \(y = x\) because \(|x| = x\); creating a line with a 45-degree slope through the origin. However, when \(x < 0\), the function evaluates to \(y = 0\), as the terms cancel each other out. This results in a flat horizontal line for negative x-values. The overall graph here is linear for positive x and horizontal for negative x, with a sharp turn or corner at the origin.

Graphs of functions can undergo transformations such as horizontal and vertical shifts. Consider the absolute value function \(y=|x+1|\). This involves a horizontal shift of the standard absolute value function, \(y=|x|\). Here, the horizontal shift moves the entire V-shape one unit to the left. The vertex of the V is thus at the point \((-1,0)\). After shifting, the graph's basic structure remains unchanged, maintaining the same slope. The general rule for horizontal shifts involves changing \(x\) to \(x+c\), shifting the graph by \(-c\) units. Vertically, a function would shift up or down based on an added value, although this example specifically concerns horizontal shifts.

A diamond-shaped graph can emerge from an absolute value equation involving two variables, like \(|x+y|=1\). This equation models a square rotated 45 degrees, centered at the origin. Rearranging the equation leads to four linear relationships: \(x+y=1\) and \(x+y=-1\), essentially linking their intercepts to four points:\((1,0)\), \((0,1)\), \((-1,0)\), and \((0,-1)\). Joining these vertices forms the diamond. This distinct form appears when the sum of absolute values of two linear expressions equals a constant, visually contrasting against lesser-known graph shapes derived from equations. The diamond's center aligns with the cross of the coordinate axes, providing a symmetrical and visually intuitive graph.