Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

What point on the plane curve represented by the parametric equations \(x=t\) and \(y=t\) corresponds to \(t=3 ?\)

Step-by-Step Solution

Answer

The point on the plane curve that corresponds to \(t = 3\) is (3, 3).

Step 1: Identify the Parametric Equations

The given parametric equations are \(x=t\) and \(y=t\). For a specific value of parameter \(t\), a point on the curve in the xy-plane can be determined by plugging that value into these two equations.

Step 2: Substitute the Given Value for Parameter \(t\)

We are given that \(t=3\). Substituting this value into the parametric equations will give the coordinates of the particular point that corresponds to \(t = 3\). So, we get \(x = 3\) and \(y = 3\) when \(t = 3\).

Step 3: Identify Point on the Plane Curve

The point on the plane curve that corresponds to \(t = 3\) is therefore (3, 3). This could be written as the ordered pair (x, y) = (3, 3).

Key Concepts

Understanding Plane Curves Through Parametric Equations

A plane curve is a continuous curve that lies on a two-dimensional plane, described using a system of equations. When we use parametric equations, the curve is delineated not by a single equation, but rather by a set of equations that define x and y coordinates in terms of one or more parameters.

In our example, the parametric equations are given by \(x=t\) and \(y=t\). This form of representation is powerful, as it allows us to describe a wider variety of curves including those that cannot be described by a function. Imagine a situation where you have to represent a looping path, like a race track; parametric equations make this possible. They provide a set of instructions on how to mentally ‘trace’ the curve if you were to draw it freehand, plotting points for each value of the parameter \(t\).

Decoding Ordered Pairs in Coordinate Systems

An ordered pair is a fundamental concept in mathematics, particularly in coordinate geometry, which comprises two elements listed in a specific order, usually as \(x, y\). These pairs represent the coordinates of a point on a plane, indicating its position relative to two intersecting lines called axes.

The first element, \(x\), is the horizontal coordinate known as the abscissa, and the second element, \(y\), is the vertical coordinate called the ordinate. By interpreting the ordered pair \(3, 3\), as described in our textbook solution, we can conclude that the point lies 3 units along the x-axis (to the right of the origin) and 3 units up the y-axis, marking it in the first quadrant of the Cartesian plane, which represents a unique location.

The Role of Coordinate Geometry in Interpreting Parametric Equations

Coordinate geometry, also known as analytic geometry, merges algebra and geometry enabling us to analyze geometric shapes using algebraic equations. In the context of parametric equations, coordinate geometry allows us to translate algebraic expressions into geometric locations on the plane.

By substituting a specific value of the parameter into the parametric equations, we can pinpoint the exact location of a point on a curve, as demonstrated in the example where we find the point corresponding to \(t=3\). This process requires a good grasp of coordinate geometry: it starts with understanding how to plot points defined by ordered pairs and extends to visualizing how these points contribute to the shape of the curve represented parametrically.

Visualizing the Curve

Consider drawing a graph and plotting multiple points by substituting different values for \(t\). Linking these individual points will reveal the visual form of the curve on the plane. Thus, coordinate geometry and parametric equations together provide a strong foundation for comprehending complex mathematical shapes and their behavior.