Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. $$\begin{aligned} &x=t^{3}+1, y=t^{3}-1, \text { for } t \text { in }[-3,3]\\\ &\text { window: }[-30,30] \text { by }[-30,30] \end{aligned}$$

Step-by-Step Solution

Answer

The rectangular equation is \(x - y = 2\).

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = t^3 + 1\) and \(y = t^3 - 1\), with the parameter \(t\) ranging from \(-3\) to \(3\). These equations represent the coordinates \((x, y)\) as functions of \(t\).

Step 2: Graph the Parametric Curves

Using a graphing calculator, input the parametric equations: \(x(t) = t^3 + 1\) and \(y(t) = t^3 - 1\). Set the parameter \(t\) to range from \(-3\) to \(3\) and adjust the graph window to \([-30,30]\) by \([-30,30]\). This will display the curve on the calculator's screen.

Step 3: Eliminate Parameter to Find Rectangular Equation

To find the rectangular equation, eliminate the parameter \(t\). From \(x = t^3 + 1\), solve for \(t^3\): \(t^3 = x - 1\). Similarly, from \(y = t^3 - 1\), solve for \(t^3\): \(t^3 = y + 1\). Set the two expressions for \(t^3\) equal to each other: \(x - 1 = y + 1\).

Step 4: Simplify to Rectangular Form

From the equation \(x - 1 = y + 1\), simplify to get \(x - y = 2\). This is the rectangular equation of the curve that represents a straight line in the \((x, y)\) plane.

Key Concepts

Parametric Equations

Parametric equations are a pair of equations that express coordinates as functions of a variable known as the parameter, often denoted by \( t \). These equations describe the path of a point in the plane over a given interval of \( t \). Instead of representing a relationship between \( x \) and \( y \) directly, parametric equations describe how both coordinates change with respect to the parameter. In this exercise, the equations \( x = t^3 + 1 \) and \( y = t^3 - 1 \) describe a curve in the \( xy \)-plane by specifying both \( x \) and \( y \) as functions of \( t \). The range of \( t \) is specified from \(-3\) to \(3\), demonstrating how the point moves along the curve.The power of parametric equations lies in their ability to describe movements and curves that would be difficult to express with a single equation. They allow for a clearer representation of paths, such as circles, ellipses, and even more complex curves, enabling us to capture motion and change over time.

Graphing Calculator

A graphing calculator is a valuable tool when working with parametric equations. It allows us to visualize the trajectory or path described by these equations easily. To use a graphing calculator for this type of exercise, you need to enter the parametric equations into their respective input fields for \( x(t) \) and \( y(t) \). Adjust the parameter range, in this case \(-3 \) to \(3\), and set the graph window, here it is specified as \([-30, 30] \) by \([-30, 30]\).With these settings, the graphing calculator will generate the path on the screen, showing how \( x \) and \( y \) change as \( t \) increases within the given interval. By visualizing the curve, students can better understand how values of \( t \) influence the coordinates and shape of the path.

Rectangular Equation

A rectangular equation is an expression that relates \( x \) and \( y \) directly, eliminating the parameter \( t \). While parametric equations provide a set of coordinates for each \( t \), the rectangular form simplifies this into a single equation. For this exercise, we need to find a connection between \( x \) and \( y \) that does not involve \( t \).Starting with the given parametric equations, we can eliminate the parameter by solving for the same expression for \( t^3 \). In this case:

• From \( x = t^3 + 1 \), we get \( t^3 = x - 1 \).
• From \( y = t^3 - 1 \), we obtain \( t^3 = y + 1 \).

By equating these two expressions, we derive \( x - 1 = y + 1 \), which simplifies to \( x - y = 2 \). This equation, \( x - y = 2 \), is the rectangular form and represents the same path in the plane as the original parametric equations, but now in the more straightforward relationship between \( x \) and \( y \).

Parameter Elimination

Parameter elimination is a process used to convert parametric equations into a rectangular form. This involves removing the parameter, \( t \), to find a direct relationship between \( x \) and \( y \). The goal is to express the curve in a simplified form, allowing for easier analysis of the path in the \(xy\)-plane.In this exercise, to eliminate the parameter, we first solve each of the parametric equations for \( t^3 \):

• Solving \( x = t^3 + 1 \) yields \( t^3 = x - 1 \).
• For \( y = t^3 - 1 \), we find \( t^3 = y + 1 \).

By setting these two expressions for \( t^3 \) equal to one another, \( x - 1 = y + 1 \), we eliminate the parameter. Simplifying gives the rectangular equation \( x - y = 2 \). This method effectively transforms the parametric form, enabling us to analyze the relationship between \( x \) and \( y \) directly, and can be particularly useful when dealing with complex motion and curves.