Problem 50 - Calculus Single Variable

Question

Describe the curve that is the graph of the given parametric equations. \(x=1 / t, y=3,0

Step-by-Step Solution

Answer

The curve is a horizontal half-line at \( y = 3 \) starting from \( x = 1 \) and extending to infinity.

Step 1: Analyze the given parametric equations

The parametric equations are given by \( x = \frac{1}{t} \) and \( y = 3 \), with \( 0 < t \leq 1 \). This indicates that for any value of \( t \) within the interval, \( y \) is constantly 3.

Step 2: Relate the parameter with the x-coordinate

Since \( x = \frac{1}{t} \), when \( t \) is at its upper bound 1, \( x = \frac{1}{1} = 1 \). As \( t \) approaches 0 from the right (since \( t > 0 \)), \( x \) approaches infinity.

Step 3: Describe the horizontal line

The equation \( y = 3 \) indicates a horizontal line at \( y = 3 \). Since \( x = \frac{1}{t} \), \( x \) varies from 1 to infinity, making it a horizontal half-line.

Key Concepts

horizontal line

A horizontal line in a coordinate plane is a straight line where all points have the same y-coordinate. This means that the line does not rise or fall as it extends left or right. Instead, it maintains a constant vertical position. In the context of the given parametric equations, the equation \( y = 3 \) represents a horizontal line at this constant y-value. Regardless of the value of \( x \), the y-coordinate remains 3.

This concept helps in visualizing how the line behaves in terms of direction. Horizontal lines have a slope of zero because there is no change in height over the change in x. When the problem states \( y = 3 \), it implies that for any value of \( x \), the line remains flat, stretching infinitely towards the left and right, or in this case, from \( x = 1 \) to infinity.

Understanding this can also support comprehension of other curves and lines, distinguishing between horizontal, vertical, and sloped lines based on varying algebraic equations.

graph of parametric equations

Parametric equations offer a versatile way to describe curves and lines by defining both x and y in terms of a third parameter, usually denoted as \( t \). This can be particularly useful for illustrating complex curves that can't be easily represented by a single function of \( x \) or \( y \).

In the given exercise, the parametric equations are \( x = \frac{1}{t} \) and \( y = 3 \), where \( 0 < t \leq 1 \). These equations describe a path or line by calculating each point from \( t \).

• As \( t \) reaches its maximum value of 1, the point on the graph is \( (x, y) = (1, 3) \).
• As \( t \) moves down towards 0, the value of \( x \) increases without bound, hence it approaches infinity while \( y \) remains at 3.

This demonstrates a horizontal line extending from a fixed point towards the right in the coordinate plane. Parametric equations thus help in charting the progression over an interval of values and showcase the path explicitly outlined by the equations of \( x \) and \( y \).

coordinate system

The coordinate system is a fundamental part of mathematics and geometry, providing a reference point for plotting points, lines, and curves. It is typically laid out on a two-dimensional plane with an x-axis that runs horizontally and a y-axis that runs vertically.

This system allows for easy representation and interpretation of mathematical equations, including parametric ones, on a graph. In the context of our parametric problem, understanding the layout helps visualize how the equation \( y = 3 \) exists as a horizontal line.

The coordinate system forms the backbone of graphing as it enables:

• Precise plotting of points - you can effortlessly locate points like \( (1, 3) \).
• Graphing lines - by following the path dictated by equations while respecting the horizontal or vertical nature of lines.
• Visualizing complex relationships - observing how curves form as each parameter, such as \( t \), varies.

Thus, mastering the coordinate system enhances one's ability to graph and interpret a wide variety of geometric and algebraic equations efficiently.