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Horizontal compression is a transformation technique used to modify the graph of a function in the x-direction. It involves shifting all the points on the graph closer to the y-axis. This gives the visual effect of squishing the graph horizontally. When we compress a graph horizontally by a factor of 4, for instance, every x-coordinate in the function is divided by 4. This makes the graph narrower as each point on the x-axis is brought closer to the y-axis.
In mathematical terms, if the original function is given by \( y = f(x) \), then the horizontally compressed function by a factor of \( c \) (where \( c > 1 \)) is given by \( y = f\left(\frac{x}{c}\right) \).
This form indicates that for every point \( (x, y) \) on the graph of \( f(x) \), the x-coordinate is divided by the compression factor. This fundamental concept is key in understanding horizontal compressions.

Function transformation is a broad concept that encompasses various operations on functions to modify their graphs. These transformations include moving (shifting), stretching, compressing, and reflecting the graph. Each operation provides a different visual change to the function's graph.
There are two main types of transformations:

• Horizontal Transformations: Affect the x-coordinates of a function. These include horizontal shifts and horizontal compressions/stretches.
• Vertical Transformations: Affect the y-coordinates, such as vertical shifts and vertical stretches/compressions.

When we deal with transformations, it is important to note whether changes affect the entire function graph or just the position/shape relative to the axes.
For example, a horizontal compression changes the distance between points along the x-axis while keeping their vertical coordinates unchanged.

The square root function is a fundamental mathematical function often represented as \( y = \sqrt{x} \). This function creates a curve that starts at the origin (0,0) and stretches towards positive infinity along both axes. It is characterized by a slow initial incline that speeds up as x increases.
For transformations like horizontal compression, understanding the nature of the square root function is crucial. The basic form, \( y = \sqrt{x} \), becomes transformed when alterations happen inside the square root.
For our example, \( y = \sqrt{x+1} \), the function is already shifted left by one unit, starting its curve at \( x = -1 \). When compressing this function horizontally by a factor of 4, we replace \( x \) with \( \frac{x}{4} \), which modifies how tightly the curve hugs the y-axis. This results in \( y = \sqrt{\frac{x}{4} + 1} \), showcasing how the base square root function is altered under such a transformation.