Problem 5 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The rectangular equation is \( y = 1 - x^2 \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = \sqrt{t} \) and \( y = 1 - t \). Here, \( t \) is the parameter. These equations represent a curve in the xy-plane, where each value of \( t \) gives a corresponding value of \( x \) and \( y \).

Step 2: Sketch the Curve

To sketch the curve, we need a few points. Let's choose values for \( t \) and compute \( x \) and \( y \).- For \( t = 0 \), \( x = \sqrt{0} = 0 \) and \( y = 1 - 0 = 1 \). Point: (0, 1)- For \( t = 1 \), \( x = \sqrt{1} = 1 \) and \( y = 1 - 1 = 0 \). Point: (1, 0)- For \( t = 4 \), \( x = \sqrt{4} = 2 \) and \( y = 1 - 4 = -3 \). Point: (2, -3)Plot these points and join them smoothly to get the curve. This curve starts at (0, 1) and passes through the points as calculated for increasing \( t \).

Step 3: Eliminate the Parameter

To eliminate the parameter \( t \), solve one of the parametric equations for \( t \) and substitute it into the other equation. From \( x = \sqrt{t} \), we have \( t = x^2 \). Substitute this into the second equation: \( y = 1 - x^2 \). This gives the rectangular-coordinate equation of the curve.

Step 4: Simplify the Resulting Equation

The resulting equation from the elimination step is \( y = 1 - x^2 \). This is a quadratic equation representing a downward-opening parabola, where the vertex is at (0, 1).

Key Concepts

Rectangular-Coordinate Equation

A rectangular-coordinate equation refers to the standard form of expressing equations using only the variables \( x \) and \( y \), which represent coordinates on a standard Cartesian plane. These equations do not involve a parameter but provide a direct relationship between \( x \) and \( y \). When we are given parametric equations, a common task is to convert them into a rectangular equation, as this can simplify analysis and visualization of the curve.For instance, in the exercise, the process of eliminating the parameter \( t \) from the parametric equations \( x = \sqrt{t} \) and \( y = 1 - t \) resulted in the rectangular equation \( y = 1 - x^2 \). This equation is much easier to interpret as it is a simple parabola equation, revealing a curve that opens downward with a vertex at point (0,1).
Understanding the transformation from parametric to rectangular form is essential for analyzing the behavior of the curve using traditional methods, such as finding intercepts and symmetry.

Parametric Curves

Parametric curves are characterized by using a parameter, such as \( t \), to express the coordinates \( x \) and \( y \) in terms of another variable. This approach provides additional flexibility in defining complex curves that might be difficult to express using traditional rectangular equations. They are extremely useful in illustrating complex motion or paths in physics and engineering.A key advantage of parametric equations is the ability to easily break down a curve into manageable parts. In the example provided, \( x = \sqrt{t} \) and \( y = 1 - t \) define a parametric curve where every specific value of \( t \) leads to a particular point \((x, y)\) on the curve. By varying \( t \) from its minimum to maximum allowable values, you can plot the curve in the coordinate plane.
This kind of plotting can help visualize the curve step-by-step, understanding how it behaves as \( t \) changes, which is often crucial in dynamic systems.

Elimination of Parameter

Eliminating the parameter in parametric equations is a process used to convert a set of parametric equations to a single rectangular-coordinate equation. This process involves solving one of the equations for the parameter and substituting it into the other equation.In the example, starting with \( x = \sqrt{t} \), we solved it for \( t \) by squaring both sides to get \( t = x^2 \). Then, we substituted \( t = x^2 \) into the second equation, \( y = 1 - t \), which resulted in the rectangular equation \( y = 1 - x^2 \).This step is crucial since it simplifies the study of the curve by providing a single, parameter-free expression. In this case, the resulting equation describes a simple parabola, allowing us to use standard techniques to analyze and graph the curve easily on a Cartesian plane. It's particularly useful for finding general features of the curve, like axes of symmetry and vertex position, without plotting individual points.