Problem 1 - Precalculus Mathematics for Calculus

Question

(a) The parametric equations \(x=f(t)\) and \(y=g(t)\) give the coordinates of a point \((x, y)=(f(t), g(t))\) for appropriate values of \(t .\) The variable \(t\) is called a _______. (b) Suppose that the parametric equations \(x=t, y=t^{2}\) \(t \geq 0,\) model the position of a moving object at time \(t\) When \(t=0,\) the object is at \((\square, \square),\) and when \(t=1\) the object is at \((\square, \square)\). (c) If we eliminate the parameter in part (b), we get the equation \(y=\) _______. We see from this equation that the path of the moving object is a _______.

Step-by-Step Solution

Answer

(a) parameter; (b) (0,0) and (1,1); (c) \(y=x^2\), parabola.

Step 1: Identify the Role of the Variable 't'

In parametric equations, the variable \( t \) serves as a parameter that defines the position \((x, y)\) of a point on a plane, based on the functions \( x = f(t) \) and \( y = g(t) \). Here, the variable \( t \) helps to map out a curve by providing coordinates for each value of \( t \). Therefore, \( t \) is called a **parameter**.

Step 2: Determine the Object's Position at Specific Times

Given the parametric equations \( x = t \) and \( y = t^2 \), we need to find the coordinates at \( t = 0 \) and \( t = 1 \). - When \( t = 0 \): - \( x = 0 \) - \( y = 0^2 = 0 \) - Thus, the position is \((0,0)\).- When \( t = 1 \): - \( x = 1 \) - \( y = 1^2 = 1 \) - Thus, the position is \((1,1)\).

Step 3: Eliminate the Parameter 't' to Find the Curve's Equation

The parametric equations \( x = t \) and \( y = t^2 \) can be combined into a single equation by expressing \( t \) in terms of \( x \). Since \( x = t \), we can substitute \( t = x \) into \( y = t^2 \), yielding the equation \( y = x^2 \). This equation represents a parabolic path.

Step 4: Identify the Type of Path

The equation \( y = x^2 \) is recognized as the standard form of a parabola. Thus, the path of the moving object is a **parabola**.

Key Concepts

Parameter

In the world of parametric equations, the variable often denoted as \( t \), plays a crucial role. This variable is known as a parameter. Why is it called a parameter? Because it acts as an independent variable that leads us to the coordinates \((x, y)\) on a plane. Imagine trying to locate a point's position in space over a period of time. The parameter \( t \) helps us do exactly that by providing specific values that correspond to the positions on the curve at different instances.

This mapping process entails:

• Using a function \( x = f(t) \) to find the x-coordinate.
• Using another function \( y = g(t) \) to determine the y-coordinate.

Each value of \( t \) represents a unique set of coordinates, allowing us to describe any kind of motion or path on a 2D plane. This makes parameters incredibly useful, particularly in modeling dynamic systems, like the path followed by a moving object.

Eliminate the parameter

Eliminating the parameter in a set of parametric equations is the process by which we convert the parametric form \((x = f(t), y = g(t))\) into a single equation that relates \( x \) and \( y \) directly. This is often desired when we want to analyze or visualize the relationship between \( x \) and \( y \) more easily.

To do this, you typically solve one of the parametric equations for \( t \), and then substitute this expression into the other equation. In our example:

• We have \( x = t \), which directly gives us \( t = x \).
• Substituting into \( y = t^2 \), we get \( y = x^2 \).

Through parameter elimination, we transformed our parametric equation into a simple and elegant function \( y = x^2 \). This transformation provides an immediate view of the relationship and helps us further understand the nature of the object’s motion in terms of \( x \) and \( y \).

Parabolic path

The equation \( y = x^2 \) reveals a very particular path, a parabolic path. A parabola is a U-shaped curve that is symmetric along its axis. In our example, as we move along the x-axis, the y-values consistently increase according to the square of the x-values. That's the beauty of a parabola—it smoothly curves upwards (or downwards, depending on the equation) from its vertex.

This kind of path:

• Is typical in trajectories, such as those seen in projectile motion.
• Can be represented by quadratic equations, signifying a squared relationship between \( x \) and \( y \).
• Has a vertex that represents the minimum or maximum point, giving it a distinctive shape.

Understanding the nature of parabolic paths is essential in fields ranging from physics to engineering, where motion or forces can be modeled as parabolas. This simple quadratic equation \( y = x^2 \) serves as a foundation for exploring more complex systems and paths.