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A horizontal shift involves moving the graph of a function left or right along the x-axis. If the transformation is in the form \(f(x-h)\), then the graph shifts horizontally. Here's what happens:

• To the right: If \(h\) is positive, meaning \(f(x-h)\), the graph will shift \(h\) units to the right.
• To the left: If \(h\) is negative, meaning \(f(x+h)\), the graph will shift \(|h|\) units to the left.

In our problem, the transformation \(f(x-3)\) indicates that every point on the graph of \(f(x)\) shifts 3 units to the right.
For example, if the original point is \((a, b)\), after the horizontal shift, the new x-coordinate becomes \(a+3\). This only affects the x-coordinate. The y-coordinate remains unchanged during a horizontal shift. It's essential to remember this distinction when analyzing graphs, as it allows us to predict the movement accurately.

A vertical shift moves a graph up or down along the y-axis. This transformation affects the y-coordinate of the points on the graph. It can be represented by \(f(x) + k\):

• Upwards: If \(k\) is positive, the graph shifts upwards by \(k\) units.
• Downwards: If \(k\) is negative, the graph shifts downwards by \(|k|\) units.

For the function \(f(x-3) + 2\), the \(+2\) indicates that the graph is shifted upwards by 2 units.
This means each point's y-coordinate will increase by 2. Given a point \((a, b)\), the vertical shift will change it to \((a, b+2)\). The x-coordinate is unaffected by this type of shift.
Understanding vertical shifts is crucial in graph transformations, as they directly affect how a function appears along the y-axis, altering its position without changing its shape.

The graph of a function visually represents the relationship between the input values (x-coordinates) and the output values (y-coordinates). Each point on the graph is a solution to the function, expressed as \((x, y)\). Transformations, such as shifting, allow us to manipulate these graphs without changing the function's fundamental nature.
When a function experiences a horizontal or vertical shift, the shape of the graph remains the same, but its position changes on the coordinate plane:

• Horizontal Shift: Moves the entire graph left or right, altering x-coordinates.
• Vertical Shift: Moves the graph up or down, changing y-coordinates.

The combined effect seen in \(f(x-3) + 2\) first moves the graph 3 units right and then 2 units up, providing a clear example of how transformations can re-position the graph while maintaining its original shape. Understanding these shifts helps in predicting the new locations of points after transformations, as seen with the original point \((a, b)\), which becomes \((a+3, b+2)\), demonstrating the impact of both the horizontal and vertical movements.