Problem 39 - Calculus Volume 3

Question

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=3 t+4 \\ y=5 t-2 \end{array} $$

Step-by-Step Solution

Answer

The equations represent a line.

Step 1: Analyze the Parametric Equations

The given parametric equations are \( x = 3t + 4 \) and \( y = 5t - 2 \). Here, \( t \) is a parameter, which commonly represents time or another varying quantity. The equations express \( x \) and \( y \) in terms of linear expressions of \( t \).

Step 2: Identify the Structure of the Equations

Both equations are linear in \( t \), meaning they are of the form \( x = at + b \) and \( y = ct + d \), where \( a \), \( b \), \( c \), and \( d \) are constants. This indicates that both \( x \) and \( y \) will vary linearly as \( t \) changes.

Step 3: Determine the Type of Curve

Parametric equations of the form \( x = at + b \) and \( y = ct + d \) typically represent a straight line, since both \( x \) and \( y \) are linear functions of \( t \). The line will pass through different points as the parameter \( t \) changes, but it maintains a constant linear relationship.

Step 4: Conclude the Type of Curve

Since both parametric equations are linear, we can conclude the set of parametric equations \( x = 3t + 4 \) and \( y = 5t - 2 \) describe a line.

Key Concepts

Linear Functions

When discussing parametric equations like \( x = 3t + 4 \) and \( y = 5t - 2 \), the concept of **linear functions** plays a critical role. Linear functions describe a straight line and can be expressed in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Here, both \( x \) and \( y \) are linear functions of the parameter \( t \).

In these parametric equations:

• \( x \) varies linearly with \( t \), where the slope is \( 3 \) and \( b \) is \( 4 \).
• \( y \) varies linearly with \( t \), with a slope of \( 5 \) and \( d \) is \(-2 \).

These linear relationships between \( x \) and \( y \) as functions of \( t \) give rise to a straight-line path in the coordinate system as \( t \) varies. As such, lines described by parametric equations maintain a consistent, direct progression, making them perfect examples of linear functions.

Curves

In mathematics, **curves** are generally continuous and smooth paths traced by points moving in relation to a parameter or set of parameters. They represent more complex relations compared to straight lines and can take many forms such as circles, ellipses, and parabolas.

Although the given exercise results in a line and not a traditional curve in the sense of wavy paths or bounded shapes, understanding line as the simplest possible curve is crucial. When parametric equations involve quadratic, circular, or other non-linear forms, the resulting paths are more obviously curved.

• For example, if the parametric representation included square terms, it could turn into a parabola.
• Circular curves might involve trigonometric functions in their parametric equations.

In this exercise, the curve traced by the equations \( x = 3t + 4 \) and \( y = 5t - 2 \) remains a straight line, considered a linear curve. However, recognizing that lines are indeed the foundational form of more complex curves helps build better intuition about advanced geometry.

Geometry

Understanding the geometry of a situation is to grasp the physical and visual representation of mathematical equations. Here, the **geometry** involved comes from interpreting the parametric equations for lines in the Cartesian plane.

Since geometry often deals with shapes, sizes, and the properties of space, translating parametric equations into geometric forms helps us see where and how these linear equations apply:

• These parametric equations define a line's pathway, giving us insight into its direction and position.
• The constants in these equations, such as \( 4 \) and \(-2 \), shift or translate the line's position within the Cartesian plane.

Visualizing these equations as geometrical paths aids in comprehending the relationship between algebraic expressions and their spatial manifestations. This turns abstract equations into tangible and understandable concepts.

Coordinate Systems

The context of **coordinate systems** is foundational for understanding how parametric equations describe paths and positions. In these systems, each point is defined by numerical coordinates, the most common being the Cartesian system.

Parametric equations like \( x = 3t + 4 \) and \( y = 5t - 2 \) essentially map the parameter \( t \) into the Cartesian plane, marking a straight path.

• The formulas convert the single dimension of \( t \) into two dimensions: \( x \) and \( y \).
• As \( t \) varies, you can plot points that trace the line, a hallmark of this coordinate system's functionality.

In coordinate systems, understanding how these parametric equations interrelate with coordinates enhances clarity on how geometric concepts like lines form and occupy space. Being able to transition between parametric forms and traditional coordinate points enables deeper insights into mathematical modeling and problem-solving.