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Reflection in function transformations is a fascinating concept where we "flip" the graph of the function over a line, such as the x-axis or y-axis. In the case of reflecting across the y-axis, each point of the function moves to the opposite side of this axis.
To reflect a function across the y-axis, you need to change the sign of the x-variable. For example, if the original function is \( f(x) = \sqrt{x} \), after reflecting across the y-axis, the new function will be \( f(-x) = \sqrt{-x} \).
In our exercise, after the function has been shifted left, the reflection requires us to adjust our modified function \( h(x) = \sqrt{x+1} \) to \( k(x) = \sqrt{-(x+1)} \). This negates both terms inside the square root, so it simplifies to \( k(x) = \sqrt{-x+1} \).
This alteration effectively mirrors all the values concerning the y-axis, flipping the graph horizontally.

A horizontal shift is used to move a function either left or right along the x-axis. This type of transformation changes the x-coordinates of the graph while keeping the y-coordinates intact.
If you want to shift a function left by a specific number of units, you must replace \( x \) with \( x + a \). Conversely, shifting right involves substituting \( x \) with \( x - a \). In this case, the value "a" determines how far and in which direction the shift will occur.

• Shifting left by 1 unit: Replace \( x \) with \( x + 1 \)
• Shifting right by 1 unit: Replace \( x \) with \( x - 1 \)

For the function \( f(x) = \sqrt{x} \), shifting it left by one unit transforms it into \( h(x) = \sqrt{x+1} \). This process ensures that each x-value moves one unit to the left on the x-axis. Thus, the graph of the function also shifts left, but retains the original shape.

Vertical shifts are a common transformation used to move a graph up or down along the y-axis. Unlike horizontal shifts, vertical shifts affect the y-coordinates of the graph while keeping the x-coordinates unchanged.
The key to performing a vertical shift is to either add or subtract a value directly to the function. Adding a value shifts the graph up, while subtracting a value shifts it down by the given number of units.

• Shifting up by 2 units: Add 2 to the function
• Shifting down by 2 units: Subtract 2 from the function

In our example, after all previous transformations, the function \( k(x) = \sqrt{-x+1} \) is shifted up by 2 units. By adding 2, we arrive at \( g(x) = \sqrt{-x+1} + 2 \). This alteration elevates every point on the graph exactly 2 units higher, modifying the vertical position but not the overall shape of the graph.