Problem 35 - Precalculus Essentials

Question

Let \(P(x, y)\) be a point on the graph of \(y=x^{2}-4 .\) Express the distance, \(d,\) from \(P\) to the origin as a function of the point's \(x\) -coordinate.

Step-by-Step Solution

Answer

The distance, \(d\), from the point \(P\) on the graph of \(y = x^{2} - 4\) to the origin as a function of its x-coordinate is \(d(x) = \sqrt{x^{2} + (x^{2} - 4)^{2}}\).

Step 1: Understand the Problem

In this problem, a point \(P\) is on the graph of the equation \(y = x^2 - 4\). The origin has coordinates (0, 0). The goal is to determine the function that gives the distance, \(d(x)\), from point \(P\) to the origin in terms of x.

Step 2: Use the Distance Formula

The distance, \(d\), between any two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) in a plane can be calculated with the distance formula: \[d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\]In this case, \(x_{1} = 0\), \(y_{1} = 0\), \(x_{2} = x\), and \(y_{2} = x^{2} - 4\), giving us:\[d = \sqrt{(x - 0)^{2} + ((x^{2} - 4) - 0)^{2}} \]

Step 3: Simplify the Equation

Simplify the equation obtained from substituting \(x_1, y_1, x_2, y_2\) to obtain \(d(x)\):\[d = \sqrt{x^{2} + (x^{2} - 4)^{2}}\]

Step 4: Express d as a function of x

The final expression for \(d(x)\) is:\[d(x) = \sqrt{x^{2} + (x^{2} - 4)^{2}}\]