91Ó°ÊÓ

Understanding a function of a variable is vital when dealing with mathematical expressions. In our exercise, the relation for the point \( (x, y) \) is defined by the equation \( y = \sqrt{x-3} \). This indicates that \( y \) is a function of \( x \). In order to switch \( x \) to a function of \( y \), we employed algebraic manipulation. By squaring both sides of the equation, we solved for \( x \) and expressed it as \( x = y^2 + 3 \). Converting between functions like this is essential when adapting formulas to different points of reference, which is a common need in geometry and calculus. This manipulation allows us to then use this expression to find the distance between points as a function of \( y \).

Square roots are an essential mathematical concept used for solving equations and understanding relationships between variables. In the function \( y = \sqrt{x-3} \), we use the square root to define \( y \) in terms of \( x \). It helps in creating a curve on a graph representing all possible points \( (x, y) \) that satisfy the equation.When understanding square roots:

• It's the inverse operation of squaring a number.
• Square roots are always non-negative in real number contexts.
• They appear often when dealing with circles, parabolas, and distance problems.

Square roots allow us to characterize the curvilinear path that a graph might take, such as a parabola opening horizontally as seen in our exercise.

Calculating the distance between two points on a graph is a fundamental skill in geometry and trigonometry. In this exercise, we use the distance formula to find the distance \( L \) between two points, \( (x, y) \) and \( (4, 0) \).The distance formula is given as:\[ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]In our exercise, this formula morphs when an expression for \( x \) in terms of \( y \) is substituted:

• First, we account for how \( x \) is expressed via \( y \): \( x = y^2 + 3 \).
• Substituting, we then derive \( L = \sqrt{1 - y^2 + y^4} \).

This result gives us the distance as a function of a single variable \( y \), simplifying the problem and allowing for easier computational analysis. It highlights how powerful algebraic manipulation is in transforming and simplifying geometrical relationships.