Problem 16 - Precalculus

Question

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=\sqrt{t+2}\\\ &y=t-1 \end{aligned}$$

Step-by-Step Solution

Answer

The sketch of the curve defined by the parametric equations should look like a half line arising from the point (0,-3) and moving in the positive direction of the x-axis. The rectangular equation equivalent to the given parametric equations is \(x = \sqrt{y + 3}\) with domain \(y ≥ -3\).

Step 1: Sketching The Parametric Equations

Start by plotting a number of points by choosing various values of \(t\), then connect these points smoothly to form the curve. For instance, when \(t= -2\), \(x= 0\) and \(y= -3\); when \(t= -1\), \(x = 1\) and \(y = -2\); when \(t = 0\), \(x = \sqrt{2}\) and \(y = -1\); and when \(t = 1\), \(x = \sqrt{3}\) and \(y = 0\). From these points, create a sketch of the graph. The orientation of the curve advances with increasing \(t\).

Step 2: Elimination Of The Parameter

Having sketched the parametric equations, now move on to the algebraic manipulation to eliminate the parameter \(t\). This can be achieved by solving one of the equations for \(t\) and then substituting it into the other equation. From the given parametric equations, it's easier to solve the second equation for \(t\), which gives \(t = y + 1\). Then, substitute \(t\) in the first equation with this solution to get \(x = \sqrt{y + 1 + 2}\) finally simplifying to \(x = \sqrt{y + 3}\).

Step 3: Adjusting The Domain

The domain adjustment should be carried out when the obtained rectangular equation is not defined for all \(x\). In this case, since \(\sqrt{y + 3}\) is only defined for \(y ≥ -3\), the domain of the rectangular equation should be adjusted accordingly.

Key Concepts

Rectangular Equations

Parametric equations describe a set of related quantities as functions of an independent parameter, often denoted as \(t\). However, sometimes it is useful to convert these parametric equations into a single equation in terms of only the variables \(x\) and \(y\). This new form is called a rectangular equation. In the given exercise, we have the parametric equations:

• \(x=\sqrt{t+2}\)
• \(y=t-1\)

To find the rectangular equation, we eliminate the parameter (in this case, \(t\)). The goal is to express \(x\) directly in terms of \(y\) (or vice versa) without \(t\). This process simplifies graphing and analysis because we work with familiar functions in one-dimensional space.

Graph Sketching

Graph sketching with parametric equations begins by selecting values of the parameter \(t\) to calculate corresponding points \((x, y)\) on the curve. By choosing a range of values for \(t\), we identify significant points to shape the curve.
For example, consider points calculated from the parametric equations:

• When \(t = -2\), \((x, y) = (0, -3)\)
• When \(t = -1\), \((x, y) = (1, -2)\)
• When \(t = 0\), \((x, y) = (\sqrt{2}, -1)\)
• When \(t = 1\), \((x, y) = (\sqrt{3}, 0)\)

Connecting these points with a smooth curve helps visualize the path, incorporating the direction, or orientation, of the curve as \(t\) increases. This technique gives a practical preview before further algebraic manipulations.

Domain Adjustment

Once a rectangular equation is obtained, adjusting the domain is crucial to ensure it only includes values that satisfy the original parametric restrictions. The domain might limit means of understanding or interpreting the graph correctly.
In our exercise, once we derived the rectangular equation \(x = \sqrt{y + 3}\), attention to domain adjustment is necessary. The expression under the square root, \(y + 3\), must be non-negative to keep the values real, hence \(y \geq -3\). This constraint narrows the range of our equation and applies directly to any graphing endeavors. Ensuring the domain reflects the true nature of the values gives a complete and accurate representation of the curve.

Parameter Elimination

Parameter elimination is a crucial step to transforming parametric equations into a rectangular form. It involves algebraic manipulation to remove the parameter \(t\) from the equations.
Typically, solve one of the parametric equations for \(t\), then substitute this expression into the other equation. In this exercise, we solve \(y = t - 1\) for \(t\), yielding \(t = y + 1\). Substituting into \(x = \sqrt{t+2}\) gives:

• \(x = \sqrt{(y + 1) + 2} = \sqrt{y + 3}\)

The result is a rectangular equation without \(t\). This step simplifies calculations and graphing, as it reduces the problem to a single function in one Cartesian coordinate system, aiding in visualization and problem-solving.