Finding Formulas for Functions


A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.

Verified step by step guidance

1. Identify the coordinates of point P on the graph of f(x) = √x. Since P lies on this graph, its coordinates can be expressed as (a, √a) for some positive value a.
2. Determine the slope of the line joining the origin (0, 0) to the point P (a, √a). The slope m is given by the formula m = (√a - 0) / (a - 0) = √a / a.
3. Express a in terms of the slope m. From the equation m = √a / a, we can rearrange to find a in terms of m: m = 1/√a, which implies √a = 1/m.
4. Solve for a by squaring both sides of the equation √a = 1/m to get a = 1/m².
5. Substitute a = 1/m² back into the coordinates of P. The x-coordinate is a = 1/m², and the y-coordinate is √a = 1/m. Therefore, the coordinates of P as functions of the slope m are (1/m², 1/m).

Key Concepts

Slope of a Line

The slope of a line measures its steepness and is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points. For a line joining the origin (0,0) to a point P(x, f(x)), the slope can be expressed as f(x)/x. Understanding this concept is crucial for relating the coordinates of point P to the slope.

Function Representation

A function represents a relationship between inputs and outputs, typically expressed as f(x). In this case, f(x) = √x defines the output for any given input x. Recognizing how to manipulate and express functions is essential for deriving the coordinates of point P in terms of the slope.

Coordinate Geometry

Coordinate geometry involves the study of geometric figures using a coordinate system. In this context, the coordinates of point P are expressed as (x, √x). Understanding how to work with coordinates in the Cartesian plane is vital for solving problems related to slopes and functions.