Problem 10 - Essential Calculus

Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=|t-1|, \quad y=t+2 $$

Step-by-Step Solution

Answer

The sketch of the curve represented by the parametric equations \( x = |t-1| \), \( y = t + 2 \) consists of two lines that intersect at the origin with orientation from the bottom left to the top right. The corresponding rectangular equation is \( y = x + 3 \) for \( x \geq 0 \), and \( y = -x + 3 \) for \( x < 0 \).

Step 1: Understand the Parameter

In the function \( x = |t-1| \) we need to understand the parameter \( t \) can be any real number. We can separate this equation into two cases: \( t < 1 \) and \( t \geq 1 \), because of the absolute value. For \( t < 1 \), \( x = 1 - t \), and for \( t \geq 1 \), \( x = t - 1 \). For the second equation \( y = t + 2 \), \( y \) increases as \( t \) increases.

Step 2: Draw the Curve

Firstly, we can draw the curve when \( t \geq 1 \). Here, as \( t \) ranges from 1 to infinity, \( x = t - 1 \) ranges from 0 to infinity. At the same time, \( y = t + 2 \) ranges from 3 to infinity. So, the curve in this case is a part of the line \( y = x + 3 \) for which \( x \geq 0 \). Secondly, for the case \( t < 1 \), as \( t \) ranges from negative infinity to 1, \( x = 1 - t \) ranges from infinity to 0, meanwhile \( y = t + 2 \) ranges from negative infinity to 3. Therefore, the curve here is a part of the line \( y = -x + 3 \) for which \( x \geq 0 \).

Step 3: Indicate the Orientation

Given that \( x \) and \( y \) are both increasing as \( t \) moves from negative infinity through to positive infinity, the orientation of the curve is from the bottom left to the top right.

Step 4: Find the Rectangular Equation

By replacing \( t \) in the \( y \)-equation with the \( x \)-equation, we can find the rectangular equation. However, this method itself has two cases. In the first case when \( t \geq 1 \), replacing \( t \) by \( x + 1 \) in the equation \( y = t + 2 \), we have \( y = x + 3 \). For the second case when \( t < 1 \), replacing \( t \) by \( 1 - x \) in the \( y \)-equation \( y = t + 2 \), we get \( y = -x + 3 \). So, due to the nature of the absolute value breaking down this relation into two pieces, the rectangular equation of \( y \) in terms of \( x \) also has two parts: \( y = x + 3 \) for \( x \geq 0 \), \( y = -x + 3 \) for \( x < 0 \).

Key Concepts

Curve Sketching

Curve sketching is an insightful way to visualize mathematical functions and the relationships they express. It involves plotting the equations' outputs to produce a graph that represents the function's behavior over a range of values. Here, we're working with parametric equations, where both coordinates of points, \( x \) and \( y \), are expressed in terms of another variable \( t \), known as the parameter.

To begin sketching the curve for this particular exercise, we use the parametric equations \( x = |t-1| \) and \( y = t+2 \). These equations suggest a dependency on the parameter \( t \), leading us to treat different cases based on values of \( t \). We have two primary cases:

• For \( t \geq 1 \), \( x = t - 1 \) results in a straightforward linear relationship as \( x \) increases, tracing a line \( y = x + 3 \) for \( x \geq 0 \).
• For \( t < 1 \), the equation modifies to \( x = 1 - t \), sketching another line \( y = -x + 3 \) for \( x \leq 0 \).

To efficiently sketch the curve in these cases, you start by identifying key points, like where \( t = 1 \). From there, observe how the line segments join along those parametric paths, creating a continuous-looking piecewise curve.

Rectangular Equations

Rectangular equations express relationships just in terms of \( x \) and \( y \), without any parameters. To obtain a rectangular equation from parametric equations—like our given \( x = |t-1| \) and \( y = t+2 \)—we eliminate the parameter \( t \). Here, the role of the absolute value function over \( t \) dictates switching the function’s expression based on whether \( t \) is greater or less than 1.

By treating the two cases separately, we decouple the parameter to form a single equation side-by-side:

• For \( t \geq 1 \), replace \( t \) by \( x + 1 \) in \( y = t + 2 \), leading to \( y = x + 3 \).
• For \( t < 1 \), substitute \( t \) with \( 1 - x \) in \( y = t + 2 \), yielding \( y = -x + 3 \).

Thus, the rectangular equation is expressed in two segments corresponding to the parameter range: \( y = x + 3 \) when \( x \geq 0 \) and \( y = -x + 3 \) when \( x < 0 \). This two-part form stems from the behavior of the absolute value, accurately capturing the parametric curve’s full trajectory.

Orientation of Curves

The orientation of a curve helps us understand the direction in which the plotted points "move" on the curve as the parameter changes. For the parametric equations \( x = |t-1| \) and \( y = t + 2 \), the parameter \( t \) sweeps across all real numbers, tracing the curve. This directional insight is especially crucial in cases involving loops, spirals, and other complex shapes.

In our exercise, as \( t \) progresses from \(-\infty\) to \(+\infty\), the curve transitions accordingly. For \( t < 1 \), the \( x \) values decrease while \( y \) values increase, suggesting movement along the segment \( y = -x + 3 \) from the upper left. As \( t \) passes \( t = 1 \), both \( x \) and \( y \) values increase, smoothly redirecting along \( y = x + 3 \) from the origin, oriented upwards and to the right.

The overall orientation thus paints a picture of a single continuous flow from one segment of the curve to the other, representing how we proceed through plotted points as initial values transform into subsequent positions. This depiction forms a trajectory from the lower left towards the upper right across the graph.