Problem 19 - 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. $$x=t^{3}+1, y=t^{3}-1 ; \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The curve is a line given by the equation \(x - y = 2\).

Step 1: Analyze the Given Parametric Equations

We are given the parametric equations:\[ x = t^3 + 1 \]\[ y = t^3 - 1 \]These equations express the coordinates \(x\) and \(y\) in terms of the parameter \(t\).

Step 2: Eliminate the Parameter

To find a rectangular equation, we need to eliminate the parameter \(t\) from the equations. Observe that:\[ x - 1 = t^3 \] and \[ y + 1 = t^3 \]Since both expressions equal \(t^3\), we can set them equal to each other:\[ x - 1 = y + 1 \]

Step 3: Simplify the Equation

Simplify the equation from the previous step:\[ x - 1 = y + 1 \]Subtract \(y\) from both sides:\[ x - y = 2 \]The rectangular equation of the curve is \(x - y = 2\).

Step 4: Graph the Rectangular Equation

The equation \(x - y = 2\) is a linear equation representing a line. It can be rewritten in the slope-intercept form as \(x = y + 2\), indicating a line with a slope of 1 and a y-intercept of -2. Graph this line on the coordinate plane to complete the representation of the curve.

Key Concepts

Rectangular Equations

Rectangular equations are a fundamental concept in algebra and coordinate geometry. They describe the relationship between the Cartesian coordinates, usually denoted by \(x\) and \(y\), without involving any parameters. This makes them efficient for graphing and understanding the shape and position of a curve in the plane. In the context of our exercise, we derived the rectangular equation \(x - y = 2\) by eliminating the parameter \(t\) from the given parametric equations. Here’s how it works: we rewrite each parametric equation in terms of \(t\) and then set the equations equal to eliminate \(t\). This results in an equation solely in terms of \(x\) and \(y\), which can then be used to graph the curve. Understanding how to switch from parametric to rectangular equations helps in visualizing the overall structure of a curve and simplifies the representation needed for graphing.

Graphing Plane Curves

When graphing plane curves, one can use rectangular equations for a straightforward visualization in a coordinate plane. With the rectangular equation \(x - y = 2\) from the exercise, we recognize it as a linear equation. By rearranging it into the slope-intercept form \(y = x - 2\), it is clear that the graph is a straight line. This can be identified easily as having:

• a slope of 1, implying it rises one unit vertically for each unit it moves horizontally
• a y-intercept at -2, meaning the line crosses the y-axis at (0, -2)

Understanding these properties allows us to plot the curve accurately on a coordinate plane. Graphing plane curves using their rectangular forms is efficient because it leverages the familiarity of line equations, enabling us to use consistent graphing methods.

Eliminating Parameters

Eliminating parameters from parametric equations is a critical step to deriving a rectangular equation. Here, the parameter \(t\) was eliminated to find a direct relationship between \(x\) and \(y\). We started with two expressions for \(t^3\):

• \(x - 1 = t^3\)
• \(y + 1 = t^3\)

By setting \(x - 1 = y + 1\), the parameter \(t\) was effectively removed. This step is crucial because it translates the parametric form, which can describe complex curves, into a rectangular form that showcases the relationship in a more standard geometrical context. Eliminating parameters showcases the power of algebraic manipulation in simplifying complex equations, making them easier to graph and understand.

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, utilizes algebraic equations to describe geometric figures on a coordinate plane. It involves understanding the placement, movement, and transformation of shapes based on equations. In this exercise, converting the parametric equations to a rectangular form is an application of coordinate geometry. The linear equation \(x - y = 2\) not only defines a line but also predicts the position and inclination of the line in a plane. Coordinate geometry combines algebra and geometry to solve problems related to shapes and curves, providing the tools to analyze and graphically represent plane figures. This bridges the gap between numerical equations and their geometric interpretations, making it a versatile branch of mathematics.