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Chapter 1: Problem 35

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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:\[d = \sqrt{(x - 0)^{2} + ((x^{2} - 4) - 0)^{2}} \]
03

Simplify the Equation

Simplify the equation obtained from substituting \(x_1, y_1, x_2, y_2\) to obtain \(d(x)\):\[d = \sqrt{x^{2} + (x^{2} - 4)^{2}}\]
04

Express d as a function of x

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

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