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Function transformations are ways to alter a function's graph to produce a new graph. These transformations can include shifts, compressions, stretches, reflections, and more. When we discuss transformations, we are working with a basic function, often called a "parent function," and altering its form. In this exercise, the parent function is the basic square root function, denoted by \( f(x) = \sqrt{x} \).

Transformations can be classified as either horizontal or vertical, depending on whether they affect the \( x \)-values or \( y \)-values, respectively. A transformation manipulates these values to modify the function's graph's position or shape. Understanding these changes is essential since it allows you to predict and understand the behavior of the transformed function. In the equation \( g(x) = \sqrt{x-9} \), a transformation has been applied to the \( x \)-values.

Graph sketching involves drawing the graph of a function based on its equation and any transformations that have been applied. For the function \( g(x) = \sqrt{x-9} \), you'll first start with the graph of the parent function \( f(x) = \sqrt{x} \).

The graph of \( f(x) = \sqrt{x} \) typically starts at the origin \((0,0)\) and curves upward and to the right. It resembles half of a parabola due to the nature of the square root. When sketching \( g(x) \), incorporate the transformation you've identified. Here, the graph will start at \((9,0)\) instead of the origin, indicating a horizontal shift.

This sketching helps visualize not only the shape but also the position of the transformed graph relative to its parent function.

Function notation is a way of expressing functions in an easily readable form, usually utilizing symbols like \( f(x) \) or \( g(x) \). In function notation, transformations are often expressed clearly in the equation. This clarity allows you to see exactly what kinds of transformations have been applied to a function.

In our example, the original parent function is \( f(x) = \sqrt{x} \). With the transformation applied, the function is expressed as \( g(x) = \sqrt{x-9} \). Using function notation, we can relate \( g(x) \) to \( f(x) \) by writing \( g(x) = f(x-9) \).

This notation indicates that each \( x \)-value of the parent function is altered by subtracting 9 before applying the square root operation. Understanding this relation is crucial in connecting parent functions to their transformed versions.

A horizontal shift involves moving the graph of a function left or right on the coordinate plane. This shift is caused by changes inside the function's argument, affecting the input \( x \).

For the function \( g(x) = \sqrt{x-9} \), the shift occurs because the parent function's input, \( x \), is replaced by \( x - 9 \). This modification results in every point on the graph moving to the right by 9 units. It is essential to remember that when the expression \( (x - c) \) appears, the shift is to the right if \( c \) is positive and to the left if \( c \) is negative.

Understanding horizontal shifts is critical in graph sketching and predicting changes to functions. They allow us to see where a graph's new position would be in comparison to its parent graph, providing insight into the transformation's effects.