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A horizontal shift occurs when a function moves left or right along the x-axis. This happens without altering the shape of the graph itself. In mathematical terms, if you shift a function horizontally by \( h \) units to the right, you substitute \( x \) in the function with \( x-h \). Conversely, for a left shift by \( h \) units, \( x \) is replaced with \( x+h \).

• To identify a horizontal shift, observe the change in x-coordinates between the original and new positions.
• If the x-value increases, the function has undergone a shift to the right.
• If it decreases, the shift is to the left.

In our exercise, the original point is (-2,2) and the shifted point is (3,4). The change in x is from -2 to 3, indicating a 5 units shift to the right. This is why \(f(x) = |x|\) is changed to \(g(x) = |x - 5| + 2\), reflecting the horizontal transition.

Vertical shifts refer to moving a graph up or down the y-axis. Similar to horizontal shifts, the shape of the function remains intact, but its position changes. When a function is shifted \( v \) units up, you simply add \( v \) to the whole function. Conversely, subtracting \( v \) shifts the graph down.

• To find a vertical shift, look at how the y-coordinate changes between points.
• A positive change means an upward shift, whereas a negative one means down.

In the given problem, the original y-coordinate 2 is changed to 4, indicating a vertical shift of 2 units upward. Therefore, after finding the horizontal shift, we add 2, modifying the function to \(g(x) = |x - 5| + 2\). This adjustment shows the function's new vertical placement.

The absolute value function, represented as \( f(x) = |x| \), is a V-shaped graph centered at the origin. It reflects the distance of x from zero, rendering it always positive or zero. Understanding the properties of absolute value is essential for applying transformations correctly.

• Absence of horizontal shifts often positions the vertex at the origin.
• Vertical transformations displace the vertex by directly altering its y-coordinate.

In the exercise, we start with \( f(x) = |x| \), and apply a horizontal shift rightwards and a vertical shift upwards with \( g(x) = |x - 5| + 2 \). Here, the vertex shifts from (0,0) in the original graph to (5,2), aligning with the calculated transformations. Absolute value functions are uniquely influenced by shifts due to their distinctive V-shape, making them a useful practice for understanding graph transformations.