Problem 12 - Precalculus with Limits

Question

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x=\sqrt{t}\) \(y=1-t\)

Step-by-Step Solution

Answer

The rectangular form of the parametric equations \(x=\sqrt{t}\), \(y=1-t\) is \(y=1-x^2\), with a domain of \(x \geq 0\). The graph of these equations is a curve which starts from the point (0,1), heading downwards to the right.

Step 1: Sketching the Curve

The parameter 't' should be considered as time, and its value can be traced back to obtain points that can be sketched on a graph (interval could be starting from 0, and increasing). The orientation is described by increasing values of 't'. Points are obtained using the equations \(x=\sqrt{t}\) and \(y=1-t\), noting that 't' ought to be a non-negative value since 'x' is the square root of 't'.

Step 2: Eliminating the Parameter

To eliminate the parameter 't', one possible method is to solve one of the equations for 't' and substitute that into the other equation. For instance, solving \(x=\sqrt{t}\) for 't' will give \(t = x^2\). Substituting \(x^2\) for 't' into \(y=1-t\) gives the rectangular equation: \(y=1-x^2\).

Step 3: Rectangular Equation Domain Adjustment

The domain will be the set of all x's for which \(y=1-x^2\) is defined. For this type of parabolic equation, all real numbers for 'x' would satisfy the equation. However, it must be remembered that due to the initial condition that 't' (and thus 'x') must be non-negative because of the square root, the domain will be adjusted to \(x \geq 0\).

Key Concepts

Rectangular Equation

A rectangular equation is a mathematical expression that relates two variables using Cartesian coordinates without any parameters. It's the familiar way of representing equations on a two-dimensional graph, like a simple line or a parabola.
For example, using the parametric equations given, we start with:

• The equation for x: \(x = \sqrt{t}\).
• The equation for y: \(y = 1 - t\).

To convert these parametric equations into a rectangular one, we need to remove the parameter \(t\). This involves solving one of the equations for \(t\) and substituting it into the other equation.
For instance, from \(x = \sqrt{t}\), squaring both sides gives us \(t = x^2\). This can then be substituted into the equation for y, \(y = 1 - t\), to yield the rectangular equation: \(y = 1 - x^2\).
This equation, \(y = 1 - x^2\), is now in the familiar form of a parabola opening downwards.

Domain Adjustment

When converting parametric equations to a rectangular form, it's essential to examine and, if necessary, adjust the domain of the resulting equation. In this context, the domain refers to all possible values of \(x\) that satisfy the equation once the parameter has been removed.
Initially, we know that in the parametric equation \(x = \sqrt{t}\), \(t\) must be non-negative because you can't take the square root of a negative number and remain in the realm of real numbers. Consequently, \(x\) itself must also be non-negative, because it represents the square root of \(t\).
This gives us a key insight: While the equation \(y = 1 - x^2\) might suggest that \(x\) could be any real number, we must restrict it to non-negative values based on the original parametric equation. Thus, our domain for \(x\) is adjusted to \(x \geq 0\).
This domain adjustment reflects the initial conditions of the problem and ensures the rectangular equation correctly represents the original parametric curve.

Eliminating Parameters

Eliminating parameters is a process of condensing parametric equations into a single equation that links the variables directly. This simplifies the representation of the curve and helps in more straightforwardly understanding the geometry of the curve on a graph.
The process involves these steps:

• Solve one of the parametric equations for the parameter. For example, from \(x = \sqrt{t}\), we can rearrange and square both sides to find \(t = x^2\).
• Substitute this expression for the parameter back into the other parametric equation. With \(t = x^2\), substituting into \(y = 1 - t\) leads to the equation \(y = 1 - x^2\).

After these calculations, the parameter is entirely removed, leaving us with a rectangular equation that describes the relationship between \(x\) and \(y\) directly.
This approach not only aids in transitioning from parametric form to rectangular but also reveals the type of curve, such as a line or parabola, represented by the equations.