Finding Formulas for Functions


Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.

Verified step by step guidance

1. Start by identifying the given point (x, y) on the graph of the function y = √(x − 3). This implies that x can be expressed in terms of y as x = y^2 + 3.
2. Recognize that the distance L between the points (x, y) and (4, 0) can be calculated using the distance formula: L = √((x - 4)^2 + (y - 0)^2).
3. Substitute the expression for x from step 1 into the distance formula: L = √(((y^2 + 3) - 4)^2 + y^2).
4. Simplify the expression inside the square root: ((y^2 + 3) - 4) simplifies to (y^2 - 1). Therefore, the expression becomes L = √((y^2 - 1)^2 + y^2).
5. Further simplify the expression: Expand (y^2 - 1)^2 to get y^4 - 2y^2 + 1. Thus, L = √(y^4 - 2y^2 + 1 + y^2), which simplifies to L = √(y^4 - y^2 + 1).

Key Concepts

Distance Formula

The distance formula calculates the distance between two points in a Cartesian plane. For points (x1, y1) and (x2, y2), the distance L is given by L = √((x2 - x1)² + (y2 - y1)²). In this problem, we will use this formula to find the distance between the point (x, y) on the curve and the point (4, 0).

Function Representation

A function represents a relationship between two variables, typically denoted as y = f(x). In this context, we need to express the distance L as a function of y, which may involve substituting y into the distance formula and rearranging the equation to isolate L. This process is essential for understanding how L varies with y.

Substitution in Functions

Substitution is a technique used in algebra and calculus to replace a variable with another expression. In this case, since y = √(x - 3), we can express x in terms of y by rearranging the equation to x = y² + 3. This substitution allows us to rewrite the distance L solely in terms of y, facilitating the analysis of the relationship between L and y.