91Ó°ÊÓ

In mathematics, a rectangular-coordinate equation is a way to represent a curve using the coordinates in the Cartesian plane, which involves the variables \( x \) and \( y \). In this form, the equation describes the relationship between these two variables without the need for an additional parameter. In our case, we started with the parametric equations \( x = |t| \) and \( y = |1 - |t|| \), where \( t \) is the parameter.To convert these parametric equations into a rectangular-coordinate equation, we need to eliminate the parameter \( t \). This means expressing \( y \) in terms of \( x \) alone. By analyzing the absolute value functions, we deduce that both When \( t \geq 0 \) and \( t < 0 \), the resulting expression for \( y \) simplifies to \( y = |1 - x| \). This rectangular equation describes the V-shaped curve that the parametric equation represents, showing how the values of \( y \) are determined solely by \( x \) on the Cartesian plane.By switching from the parametric form to a rectangular-coordinate equation, we better understand the global shape and behavior of the curve, making it easier to graph and study without reference to a parameter like \( t \).

Curve sketching is a fundamental skill that helps us visualize mathematical equations by drawing their respective graphs. The goal is to understand the shape and behavior of the curve by analyzing key features and specific points. For the given parametric equations \( x = |t| \) and \( y = |1 - |t|| \), sketching the curve involves identifying how \( x \) and \( y \) change with respect to the parameter \( t \).Looking at the parametric values:- When \( t = 0 \), we find \( x = 0 \) and \( y = 1 \), which gives us the point (0, 1).- For \( t = 1 \) or \( t = -1 \), \( x = 1 \) while \( y = 0 \), resulting in two points: (1, 0) and (1, 0).- As \( |t| \rightarrow \infty \), \( x \) increases indefinitely, and \( y \) approaches 1.By plotting these points, we notice a V-like shape on the graph. The curve is symmetric about the y-axis and forms a continuous path as \( t \) varies. This sketching process offers a visual representation of how the curve extends in the plane, giving insights into intercepts, asymptotic behavior, and overall geometry without having to calculate every possible value.

The absolute value function is essential for the given problem because both parametric equations involve this concept. An absolute value function, represented as \( |x| \), returns the non-negative value of \( x \), disregarding its sign. It's especially vital when you're seeking to eliminate or explore the effects of specific parameters or conditions across potential ranges.For the equation \( x = |t| \), \( x \) remains non-negative for any real number \( t \), meaning it captures the magnitude of \( t \). Similarly, \( y = |1 - |t|| \) computes the absolute difference between 1 and \( |t| \), which signifies the distance between \(|t|\) and 1. This function ensures that the results for both \( x \) and \( y \) are non-negative, preserving the graph's integrity by treating negative inputs differently.Understanding how absolute value impacts a function helps visualize and interpret the graph's symmetry and key points. In our scenario, absolute value is instrumental in shaping a symmetric V-curve around the y-axis, highlighting how small changes in \( t \) can lead to larger variations in the curve's structure and direction.