Problem 36 - Precalculus Essentials

Question

Let \(P(x, y)\) be a point on the graph of \(y=x^{2}-8 .\) 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 \(P\) to the origin as a function of the point's \(x\) -coordinate is \(\sqrt{x^{4}-15x^{2}+64}\).

Step 1: Define the function

The question provides a function \(P(x, y)\) that belongs on the graph of \(y=x^{2}-8\). We can simplify things by considering \(y\) as a function of \(x\), therefore \(y = f(x) = x^{2}-8\).

Step 2: Apply the distance formula

The distance \(d\) from any point \((x, y)\) to the origin \((0, 0)\) is calculated using the distance formula \(d=\sqrt{x^{2}+y^{2}}\). As we have defined \(y = f(x) = x^{2}-8\), we can replace \(y\) in the distance formula with \(x^{2}-8\). This gives us: \(d=\sqrt{x^{2}+(x^{2}-8)^{2}}\).

Step 3: Simplify the equation

Now, use algebraic rules to reduce the distance formula equation. This results in: \(d = \sqrt{x^{2} + x^{4} - 16x^{2} + 64} = \sqrt{x^{4}-15x^{2}+64}\)