Problem 24 - A Graphical Approach to Precalculus with Limits

Question

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. $$x=t^{2}+2, y=t^{2}-4 ; \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \(x - y = 6\).

Step 1: Graph the Parametric Equations

To graph the curve given in parametric form \(x = t^2 + 2\) and \(y = t^2 - 4\), consider several values for \(t\). For example, if \(t = 0\), \(x = 2\) and \(y = -4\). If \(t = 1\), \(x = 3\) and \(y = -3\), and so on. By plotting these points and others within a reasonable range for \(t\), you'll notice the path traced by the curve on the \(xy\)-plane.

Step 2: Eliminate the Parameter

To find a rectangular (Cartesian) equation for the curve, eliminate the parameter \(t\). Start by solving for \(t^2\) from one of the parametric equations. From \(x = t^2 + 2\), we get \(t^2 = x - 2\). Similarly, from \(y = t^2 - 4\), we get \(t^2 = y + 4\).

Step 3: Establish the Relationship Between x and y

Since both equations from Step 2 equal \(t^2\), we can set them equal to each other:\(x - 2 = y + 4\) . Simplify this equation to get \(x - y = 6\). This is the rectangular equation of the curve.

Key Concepts

Graphing Plane Curves

Graphing plane curves involves plotting a set of equations that describe both x and y coordinates in terms of a third variable, known as a parameter, often represented as \(t\). This is different from the typical Cartesian coordinates, where you might have y expressed directly in terms of x.
To graph these parametric equations, like \(x = t^2 + 2\) and \(y = t^2 - 4\), you need to substitute various values of \(t\) to find corresponding x and y points. For instance, if \(t = 0\), \(x = 2\) and \(y = -4\) gives you a point (2, -4) on the plane.

• Substitute different values of \(t\), like \(t = 1\) gives \(x = 3\) and \(y = -3\).
• Continue with more values to see the trend, forming a set of points (x, y) to plot.
• Recognize the shape and direction of the curve as you plot more points.

By plotting enough points, the distinct pattern or shape formed by the curve becomes apparent, offering valuable insights into its behavior.

Rectangular Equations

Rectangular equations, also known as Cartesian equations, describe a curve in the plane using only x and y coordinates. This format removes the parameter \(t\) that was originally involved in parametric equations.
Converting to a rectangular equation often makes it simpler to analyze the relation between x and y directly. Once converted, the equation lets us understand the curve's characteristics, like its slope, intercepts, and any symmetries or patterns.
For the given parametric equations \(x = t^2 + 2\) and \(y = t^2 - 4\), our goal is to express the relationship between x and y without including \(t\).

• Solve for \(t^2\) in terms of x: \(t^2 = x - 2\).
• Solve similarly for y: \(t^2 = y + 4\).
• Set these equal due to their common value in terms of \(t^2\), which gives \(x - 2 = y + 4\).

This result, \(x - y = 6\), is a straight line when plotted, showing that the parameterized curve indeed translates to a simple linear relationship in Cartesian coordinates.

Parametric to Cartesian Conversion

Converting parametric equations to Cartesian equations is a crucial skill in analyzing curves without involving the parameter \(t\). This process simplifies the study of curves by providing a single equation relating x and y variables directly.
To carry out this conversion effectively, start by isolating \(t\) or its expressions involving powers, like \(t^2\), from one of the parametric equations. For instance, from \(x = t^2 + 2\), derive \(t^2 = x - 2\).
Do the same for the other equation, \(y = t^2 - 4\), obtaining \(t^2 = y + 4\). These equalities imply the parameter is eliminated when the two expressions for \(t^2\) are set equal:

• Equate \(x - 2 = y + 4\), which results in \(x - y = 6\).
• This direct relationship between x and y allows for straightforward graphing and analysis without the extra step of parameter substitution.

By eliminating the parameter, you streamline your understanding of the curve, transitioning from a parametric perspective to a simpler Cartesian equation.