Problem 326 - Calculus Volume 3

Question

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=1+t, \quad y=t^{2}-1, \quad-1 \leq t \leq 1 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( y = x^2 - 2x \) for \( 0 \leq x \leq 2 \) and \(-1 \leq y \leq 0 \).

Step 1: Understand the Parametric Equations

We have two parametric equations: \( x = 1 + t \) and \( y = t^2 - 1 \). These equations use \( t \) as a parameter which defines the points \( (x, y) \) on the curve. Our goal is to describe this curve in Cartesian coordinates by eliminating \( t \).

Step 2: Solve for Parameter \( t \)

From \( x = 1 + t \), we solve for \( t \) to get \( t = x - 1 \). This expression will help us eliminate \( t \) from the equation for \( y \).

Step 3: Substitute to Find \( y \) in Terms of \( x \)

Substitute \( t = x - 1 \) into the second equation \( y = t^2 - 1 \). This gives us \( y = (x - 1)^2 - 1 \).

Step 4: Expand and Simplify the Cartesian Equation

Expand \( (x - 1)^2 \) to get \( x^2 - 2x + 1 \). Substitute back to get the equation \( y = x^2 - 2x + 1 - 1 = x^2 - 2x \). This is the Cartesian equation of the curve.

Step 5: Determine the Range of \( x \) and \( y \)

Since \(-1 \leq t \leq 1\), substituting into \( x = 1 + t \) gives the range \( 0 \leq x \leq 2 \). Similarly, substituting into \( y = t^2 - 1 \), we find \(-1 \leq y \leq 0\) when \( t \) is at its limits.

Step 6: Sketch the Curve

Plot the points on the graph for \( 0 \leq x \leq 2 \) and \(-1 \leq y \leq 0 \). Since the equation is a downward-opening parabola (part of \( y = x^2 - 2x \)), draw the curve between these x and y ranges.

Key Concepts

Cartesian Equation

The Cartesian equation is derived from parametric equations by removing the parameter. It represents the relationship between the x and y coordinates directly, which is easier to plot and analyze.

In our exercise, the parametric equations are given as:

• \( x = 1 + t \)
• \( y = t^2 - 1 \)

We aim to eliminate the parameter \( t \) to express \( y \) purely in terms of \( x \).

This process involves solving one of the parametric equations for \( t \) and substituting it into the other equation. Once \( t = x - 1 \) is substituted into \( y = t^2 - 1 \), the result is a Cartesian equation:

• \( y = (x - 1)^2 - 1 \)
• Expanding the expression gives: \( y = x^2 - 2x \), which is a simplified quadratic equation.

This equation highlights the symmetrical, parabolic shape typical in quadratic graphs.

Parameter Elimination

Parameter elimination is a crucial step in transitioning from parametric to Cartesian form. It involves finding a way to express the common parameter \( t \) in terms of either \( x \) or \( y \), allowing it to be removed.

For our exercise, we started with \( x = 1 + t \). By rearranging it, we obtained \( t = x - 1 \). With \( t \) expressed in terms of \( x \), this expression is substituted into the equation for \( y \).

• Original: \( y = t^2 - 1 \)
• Substituted: \( y = (x - 1)^2 - 1 \)

Essentially, we have 'eliminated' \( t \) from these equations, providing a clear view of the curve's behavior solely in relation to \( x \) and \( y \). This step is often simplified using algebraic manipulation and square expansions.

Range of x and y

Determining the range of \( x \) and \( y \) is essential for understanding the limits and shape of the curve.

In this exercise, the range of \( t \) is given:

• \(-1 \leq t \leq 1\)

Using \( x = 1 + t \), this transforms into the range

• \( 0 \leq x \leq 2 \)

Similarly, by plugging the range of \( t \) into \( y = t^2 - 1 \), we determine the range for \( y \) as:

• \(-1 \leq y \leq 0\)

These ranges tell us that the part of the parabola we are considering is restricted in a very specific area of the graph. Understanding these constraints ensures we accurately plot only the relevant portion of the curve.

Curve Sketching

Sketching the curve from a Cartesian equation involves knowing the general shape it takes and plotting relevant points based on the range of \( x \) and \( y \).Understanding that our equation, \( y = x^2 - 2x \), forms a downward-opening parabola, helps in predicting its shape.

Important points include the vertex and the intersection points on the \( x \)-axis or limits of \( y \).

• Since the range of \( x \) is \( 0 \leq x \leq 2 \), plot this interval on the x-axis.
• Within these limits, sketch the parabola from \( (0, -1) \) to \( (2, -1) \), maintaining its opened downward form.

This visualization provides a better understanding of the behavior of the curve over its defined intervals, crucial for comprehensively interpreting any parametric-determined graph.