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Understanding the distance formula is crucial when working with geometric problems. The distance formula allows us to calculate the distance between two points in a coordinate plane. It's derived from the Pythagorean theorem. For points o point 1:

• (x鈧�, y鈧�)

and point 2:

• (x鈧�, y鈧�)

,the formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] In this exercise, we are calculating the distance between points

• (x, y)

and

• (4, 0)

. By substituting these values into the distance formula, we obtain: \[ L = \sqrt{(x-4)^2 + y^2} \] Understanding this formula helps us transition to expressing the distance as a function of another variable. We leverage relationships between the coordinates to simplify the problem.

Graphs are visual representations of mathematical functions, and they help us understand relationships between variables. In this exercise, we consider the graph of the function \(y = \sqrt{x-3}\). This is a square root function, which typically forms a curve that starts at a certain point and. Then gently rises. The point

• (4,0)

also lies on the coordinate plane, where the graph of

• \(y = \sqrt{x-3}\)

intersects or relates to other points. To represent this relationship on the graph, we can express \(x\) in terms of \(y\), leading to \(x = y^2 + 3\). This substitution allows us to manipulate and understand the graph with respect to specific values of \(y\). Graphs help illustrate complex algebraic relationships and provide insight into the behavior of the function over a range of values.

Algebraic manipulation involves rearranging and simplifying mathematical expressions to reveal new insights or solve problems. In this exercise, we use algebraic skills to express the distance \(L\) as a function of \(y\). First, knowing

• \(y = \sqrt{x-3}\)

, we rearrange it to find \(x\): \[x = y^2 + 3\]. Next, substitute this expression for \(x\) in the distance formula: \[ L = \sqrt{((y^2 + 3) - 4)^2 + y^2} \].Now, simplify the expression. Inside the square root:

• \((y^2 + 3 - 4)^2 = (y^2 - 1)^2\)

. Continuing further gives us \[ L = \sqrt{(y^2 - 1)^2 + y^2} \].These steps demonstrate how the process of algebraic manipulation helps break down and solve complex expressions. This tool is essential for any mathematical task, simplifying expressions, solving equations, or finding unknowns.

A square root function is a fundamental mathematical concept represented as \(y = \sqrt{x}\). In this exercise, we specifically work with a shifted square root function, \(y = \sqrt{x-3}\). Square root functions often start at a certain point on the x-axis and curve upwards gently to the right. This behavior characterizes the curve and its progression as \(x\) increases.Square root functions have certain properties:

• They are only defined for non-negative values inside the square root (\(x - 3 \geq 0\); here, \(x \geq 3\)).
• Their range is also non-negative values taking values from 0 onwards.
• They are not linear and do not cross the x-axis or y-axis.

Understanding the nature of the square root function assists in effectively analyzing and graphing these curves to explore their properties further. Such exploration develops a deep comprehension of the interdependency between variables. The behavior of square root functions is critical in both pure and applied mathematical settings.