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An absolute value function is a type of function that measures the distance of a number from zero on a number line. It's often represented as \( y = |x| \). This means, for any given \( x \), \( |x| \) will always result in a non-negative value. It's like asking, "How far is this number from zero?" whether it's positive or negative.
For example:

• If \( x = 3 \), then \( |3| = 3 \) because 3 is already a positive number.
• If \( x = -3 \), then \( |-3| = 3 \) because the distance from -3 to 0 is also 3, converting the negative to positive.

In graphing, the absolute value function typically produces a "V" shape, centered at the origin (0,0) if no other transformations are applied.

Function transformation is a process of changing a graph in certain ways, such as moving it up, down, left, or right. Transformations can also include changes in size or orientation. For the function \( y = |x| + 1 \), the "+1" signifies a vertical transformation.
This transformation shifts the entire graph of \( y = |x| \) up by one unit. To visualize this, imagine each point on the graph of \( y = |x| \) moving one step higher on the y-axis, maintaining its V shape but with the vertex now at (0,1) instead of the origin. This is an example of a vertical shift, which works to move all points of a graph directly up or down depending on the sign and magnitude of the added constant.

Coordinate plotting is the process of placing points on a graph using pairs of numbers, called coordinates, which represent locations on the Cartesian plane. A coordinate is written in the form \((x, y)\) where \( x \) indicates the horizontal position, and \( y \) indicates the vertical position.
For the equation \( y = |x| + 1 \), you can form coordinates by choosing values for \( x \) and calculating \( y \) using the equation. Here's how you would calculate the coordinates you used:

• For \( x = -3 \), substitute into the equation \( y = |-3| + 1 = 4 \), plotting the point \((-3, 4)\).
• For \( x = 0 \), substitute into the equation \( y = |0| + 1 = 1 \), plotting the point \((0, 1)\).
• For \( x = 1 \), substitute into the equation \( y = |1| + 1 = 2 \), plotting the point \((1, 2)\).
• The same method applies to the other x-values: -2, -1, 2, and 3.

Once these points are plotted, you connect them to visually form the graph of the transformed absolute value function, which is a V shape that appears shifted up.