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A horizontal shift in the graph of a function involves moving the whole graph to the left or to the right. In the context of the function transformation provided, the expression \( f(x+2) \) denoted that every point of the original function \( f(x) \) is shifted horizontally. This concept can be a bit tricky to remember because the direction seems counterintuitive:

• Replacing \( x \) with \( x + c \) in the function results in a shift to the left by \( c \) units.
• On the other hand, \( f(x - c) \) implies a shift to the right by \( c \) units.

To find out where each point on the graph of \( y = f(x) \) has moved, take every \( x \)-coordinate and adjust it by the specific number of units the function is shifted. In this exercise, the original points at \( (0, 1), (1, 2), \) and \( (2, 3) \) become \( (-2, 1), (-1, 2), \) and \( (0, 3) \) respectively after a horizontal shift of 2 units to the left. As seen here, the horizontal shift only affects the \( x \)-values, making each decrease by 2.

The vertical shift concept involves adjusting the graph of a function up or down along the \( y \)-axis. In the transformation of \( y = f(x+2) - 1 \), the \( -1 \) signifies a vertical shift downwards.

• To move a graph **down**, simply subtract the desired value from \( y \) in the function (\( f(x) - c \)).
• Conversely, adding a value results in an upward shift (\( f(x) + c \)).

This means that each \( y \)-coordinate of the points of the original function \( y = f(x) \) decreases by 1 unit. As an effect of this shift, each of the \( y \)-coordinates of the transformed points was adjusted one unit lower, resulting in the points \( (-2, 0), (-1, 1), \) and \( (0, 2) \). This transformation effectively repositions the entire graph without altering its shape or orientation, similarly to how we might pull down a shade on a window.

Coordinate transformation refers to the systematic alteration of the coordinates of points on a graph based on certain mathematical rules or operations. With both horizontal and vertical shifts being applied, a coordinate transformation results in a change in both \( x \)-coordinates and \( y \)-coordinates.

• The resulting transformation is an overlapping effect where each point's original position is altered both horizontally (left/right) and vertically (up/down).
• This is a two-step process, where the horizontal shift is usually applied first, followed by the vertical shift. However, these can occur simultaneously for each coordinate point.

For the provided exercise, the coordinate transformation turned the original points at \( (0, 1), (1, 2), \) and \( (2, 3) \) into \( (-2, 0), (-1, 1), \) and \( (0, 2) \) respectively. With each point, we initially shifted horizontally by subtracting 2 from the \( x \)-coordinate, and then vertically by subtracting 1 from the \( y \)-coordinate, leading to new positions on the graph. This systematic shift redefines the function's appearance without distorting its fundamental properties or how it behaves.