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The square root function is a fundamental mathematical function represented by the square root symbol, \( y = \sqrt{x} \). In our specific example, the function is \( y = \sqrt{x-7} \), which signifies that the square root is applied to the value \( x-7 \). This type of function is essential because it represents a relationship where the output is the principal (or non-negative) square root of the adjusted input. When graphing a basic square root function, the graph typically starts at the origin \((0, 0)\) and creates a curve that extends to the right, resembling an arc or a half-parabola. This smooth, increasing curve continues to climb slowly as \( x \) increases.

Understanding the domain and range of a function is crucial when graphing it or analyzing its behavior.
- **Domain** refers to all the possible input values (\( x \)) for the function. For \( y = \sqrt{x-7} \), it's important to only consider values of \( x \) that make the expression under the square root non-negative. This results in the inequality \( x-7 \geq 0 \), meaning that \( x \geq 7 \). Thus, the domain is \([7, \infty)\).
- **Range** refers to all possible output values (\( y \)) that the function can produce. Since the square root function outputs non-negative numbers, starting from 0 when \( x = 7 \), the range for \( y = \sqrt{x-7} \) is \([0, \infty)\). These two concepts help in sketching and understanding the full behavior of the function on a graph.

An x-axis shift is a transformation that moves a function horizontally from its original position on the graph. This can either be to the left or the right, depending on the expression inside the function.
In \( y = \sqrt{x-7} \), the \( -7 \) inside the square root causes a shift of 7 units to the right along the x-axis. This is because the function's starting point (or vertex) shifts from \( x = 0 \) to \( x = 7 \). This horizontal shift is a common strategy for modifying where the graph of a function begins. It does not affect the shape of the graph, just its starting position along the x-axis.

Graphing functions involves plotting several points on the coordinate plane and then connecting these points to reveal the overall shape of the graph. For \( y = \sqrt{x-7} \), begin by selecting values of \( x \) from the domain, such as 7, 8, 9, and so on.

• At \( x = 7 \), \( y = \sqrt{7-7} = 0 \).
• At \( x = 8 \), \( y = \sqrt{8-7} = 1 \).
• At \( x = 9 \), \( y = \sqrt{9-7} = \sqrt{2} \approx 1.41 \).

After calculating several such values, plot these points on a graph. Connect these points to form a smooth curve that represents the square root function. This curve, starting at (7, 0), extends to the right, always remaining above the x-axis and never dropping below it. This graphical representation gives a clear visual understanding of how the function behaves and emphasizes the curve's progression as \( x \) increases.