Problem 38 - Precalculus Essentials

Question

Let \(P(x, y)\) be a point on the graph of \(y=\sqrt{x} .\) Express the distance, \(d,\) from \(P\) to (2,0) as a function of the point's \(x\) -coordinate.

Step-by-Step Solution

Answer

The distance, \(d\), from point \(P\) to (2,0), expressed as a function of the point's \(x\)-coordinate, is \(d = \sqrt{(x-2)^2 + x}\).

Step 1: Understand the distance formula

The distance formula in a two-dimensional space is given by: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)where the points are \((x_1, y_1)\) and \((x_2, y_2)\) respectively.

Step 2: Identify the points

The points are \(P(x, y)\) and (2,0). On the function \(y = \sqrt{x}\), \(y\) can be written as \(\sqrt{x}\). Therefore, we will substitute \(\sqrt{x}\) for \(y\). The points become \(P(x, \sqrt{x})\) and (2,0).

Step 3: Substitute into the distance formula

Let's substitute into the distance formula:So we have: \(d = \sqrt{(x-2)^2 + (\sqrt{x}-0)^2} = \sqrt{(x-2)^2 + x}\)This is the function of \(d\) with respect to \(x\).

Key Concepts

Distance Between Points

When we encounter a problem involving the distance between two points, it's crucial to understand the tool that allows us to calculate this measurement: the distance formula. Imagine you're walking from one location to another; the distance formula represents the straight-line path you'd take, as if a bird flew from one point to the other.

The distance formula is derived from the Pythagorean Theorem, which is used to find the length of the hypotenuse of a right triangle. For this reason, it's often visualized as the diagonal line of a right-angled triangle formed between the two points on a graph. In its most common form, the formula is
\[\begin{equation}d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\end{equation}\]
where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the points,
• \(d\) is the distance between these points.

By applying the distance formula, we can understand the relationship between any two locations on the coordinate plane, making it an essential tool not just in precalculus, but in various applications like navigation, physics, and computer graphics.

Square Root Function

The square root function is an important mathematical concept, represented as \(y = \sqrt{x}\). It gives us the principal square root of any nonnegative number \(x\). This function is actually the inverse of the function \(y = x^2\) for \(x \ge 0\). When you graph this function, it produces a curve that starts at the origin (0,0) and moves upward to the right in a smooth, continuous way.

In the context of distance calculations, using the square root is essential when we apply the distance formula. Any time you see a square root appear, remember that it is undoing a square and finding the original value that was multiplied by itself. Visually, this can be imagined as the measure of the side of a square that has an area equal to \(x\). Understanding how to manipulate and work with the square root function is key, as it can transform equations and relationships between variables in algebraic problems.

Function Notation

Understanding function notation is like learning the language that math uses to describe relationships between variables. It is a concise way to show how a particular input, usually represented by \(x\), leads to a specific output, named \(y\). For example, if we have a function that adds 5 to any input, we could write it as \(f(x) = x + 5\), where \(f\) is the name of the function.

In the exercise provided, the goal is to express the distance \(d\) as a function of \(x\), which requires us to use function notation. Here, instead of y, we're focusing on \(d\), the distance. After finding the expression through the distance formula involving \(x\), we could denote it in function notation as \(d(x)\). This means that \(d\) is a function that depends on the value of \(x\). Every time \(x\) changes, \(d\) will change too, and that's precisely the heart of function notation - to capture the essence of this dynamic relationship in a simple and consistent format.