Problem 25 - A Graphical Approach to College Algebra

Question

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t, y=\sqrt{t^{2}+2}, \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

Rectangular equation: \( y = \sqrt{x^2 + 2} \), with \( x \) in \((-\infty, \infty)\).

Step 1: Identify the Parametric Equations

The given parametric equations are \( x = t \) and \( y = \sqrt{t^2 + 2} \). These equations define the path of a point \((x, y)\) on the curve as the parameter \( t \) varies over its interval.

Step 2: Express \( t \) in terms of \( x \)

Since \( x = t \), we can directly substitute \( t \) with \( x \). This gives us \( t = x \).

Step 3: Substitute \( t = x \) into the equation for \( y \)

Replace \( t \) in the equation for \( y \) with \( x \): \[ y = \sqrt{x^2 + 2} \].This is the rectangular equation of the curve.

Step 4: Determine the Interval for \( x \)

Since \( t \) spans the interval \((-\infty, \infty)\), and \( t = x \), the interval for \( x \) is also \((-\infty, \infty)\).

Key Concepts

Parametric Equations

Parametric equations allow us to express the coordinates of points on a curve using a third variable, typically denoted as \( t \). This approach is incredibly useful, especially for curves that are not easy to describe using a single equation in rectangular form. In this example:

• We have two parametric equations: \( x = t \) and \( y = \sqrt{t^2 + 2} \).
• These equations describe the curve indirectly by letting \( t \) vary across a range, thereby tracing out the curve point by point.

Parametric equations provide flexibility. Instead of using traditional \( y = f(x) \), we separate the motion or path into components, \( x(t) \) and \( y(t) \), giving a dynamic view of curves, especially useful in physics and engineering.

Interval Notation

Interval notation is a way to describe the set of numbers that a variable can take. It's written with parentheses or brackets:

• Parentheses \((a, b)\) denote that \( a \) and \( b \) are not included (open interval).
• Brackets \([a, b]\) mean \( a \) and \( b \) are included (closed interval).

In the exercise, the interval for \( t \) is \((-\infty, \infty)\), meaning \( t \) can take any real value. This translates directly to \( x \) because \( x = t \). Thus, the interval for \( x \) is also \((-\infty, \infty)\).
Interval notation is helpful to specify the domain, making it clear which values are included. It’s a concise and powerful tool for describing the constraints or range of variables in equations.

Curve Transformation

Curve transformation involves changing a curve's representation to another form. With parametric equations, this often means converting so-called parametric form into a rectangular equation, which is a more familiar setup for many.

• To transform, first express \( t \) in terms of a simpler variable, like finding \( t = x \) here.
• Next, substitute this expression into the other parametric equation to eliminate \( t \).

In this case, substituting \( t = x \) into \( y = \sqrt{t^2 + 2} \) gives the rectangular form \( y = \sqrt{x^2 + 2} \).
This transformation clarifies the relationship between \( x \) and \( y \), allowing for easier graphing and analysis. Curve transformations are key in mathematics as they simplify problems and link various forms of representation.