Problem 5 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\sqrt{t}, \quad y=1-t $$

Step-by-Step Solution

Answer

The curve is a downward-opening parabola represented by the equation \(y = 1 - x^2\) for \(x \geq 0\).

Step 1: Understanding Parametric Equations

Parametric equations express a set of related quantities as explicit functions of an independent parameter. Here, we have the parametric equations: \( x = \sqrt{t} \) and \( y = 1 - t \). We need to sketch this curve and find the corresponding rectangular (Cartesian) equation.

Step 2: Sketching the Curve

To sketch the curve, determine how \(x\) and \(y\) change with \(t\). For \(t \geq 0\), \(x\) is the square root function, and it increases as \(t\) increases. Meanwhile, \(y\) decreases linearly. Plot a few points: if \(t = 0\), then \(x = 0\) and \(y = 1\); if \(t = 1\), then \(x = 1\) and \(y = 0\). This gives a curve starting at (0,1) descending right to (1,0).

Step 3: Eliminating the Parameter

To eliminate the parameter \(t\), solve one of the parametric equations for \(t\). From \(x = \sqrt{t}\), we have \(t = x^2\). Substitute \(t = x^2\) into the equation for \(y\): \(y = 1 - t = 1 - x^2\). The rectangular equation for the curve is \(y = 1 - x^2\).

Step 4: Analyzing the Rectangular Equation

The rectangular equation \(y = 1 - x^2\) represents a downward-opening parabola with vertex at (0,1). This matches our sketch from the parametric equations. The domain of the original parametric curve \(t \geq 0\) implies \(x \geq 0\) for the parabola.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a method of graphing points and curves in a plane using two axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this system is defined by a pair of numbers, \(x, y\), which represent its distance from the axes. By applying this system, we can transform parametric equations into more familiar forms that are easier to analyze.
In our exercise, once we eliminate the parameter and obtain the rectangular equation \(y = 1 - x^2\), we position this equation in the rectangular coordinate system. Now we can quickly recognize it as a parabola, providing a clear visual representation of the relationship between \(x\) and \(y\).
This system allows for a consistent and uniform means to visualize mathematical relationships, bridging the understanding between abstract parametric forms and their geometric interpretations.

Curve Sketching

Curve sketching involves drawing a graph of an equation to show its general shape in a coordinate system. When dealing with parametric equations, understanding how each variable changes with the parameter is crucial.
By examining the parametric equations \(x = \sqrt{t}\) and \(y = 1 - t\), we discern that as \(t\) increases, \(x\) grows while \(y\) decreases. Plotting individual points by assigning different values to \(t\), such as \(t = 0\) resulting in \(x = 0\), \(y = 1\) and \(t = 1\) yielding \(x = 1\), \(y = 0\), assists in sketching.
The resulting curve is a segment of a downward-sloping line starting at \( (0, 1) \) and moving towards \( (1, 0) \). This visualization process is essential for grasping the parametric relationship and its graphical form, enabling a deeper insight into the behavior of equations and functions.

Parameter Elimination

Parameter elimination is a process used to remove the parameter from a set of parametric equations, resulting in a standard equation in the rectangular coordinate system. This is typically achieved by solving one of the parametric equations for the parameter itself.
In this exercise, from the equation \(x = \sqrt{t}\), we solve for \(t\) to get \(t = x^2\). We then substitute this back into the equation for \(y\), transforming it into a rectangular equation: \(y = 1 - x^2\).
This newly obtained equation represents the same curve as the original parametric equations without explicitly involving the parameter \(t\). The elimination of the parameter not only simplifies the equations but also enhances our ability to interpret the overall structure of the curve as it relates to familiar shapes such as lines and parabolas.
By converting parametric equations into their rectangular form, we facilitate an enhanced understanding of the curve’s geometric properties and its behavior across the rectangular coordinate plane.