Problem 10 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=|t|, \quad y=|1-| t| | $$

Step-by-Step Solution

Answer

The curve is a V-shape and its rectangular equation is \( y = |1 - x| \).

Step 1: Analyze the Parametric Equations

The given parametric equations are \( x = |t| \) and \( y = |1 - |t|| \). These represent how \( x \) and \( y \) depend on the parameter \( t \). The domain for \( t \) is all real numbers, and hence \( x \) and \( y \) are defined for all \( t \).

Step 2: Understanding the Range of x and y

Since \( x = |t| \), \( x \) will always be non-negative, meaning \( x \geq 0 \). For \( y = |1 - |t|| \), \( y \) will be the distance from \( |t| \) to 1. Thus, \( y \geq 0 \) and \( y \) will be 1 when \( |t| = 0 \), and will increase to 1 as \(|t|\) approaches infinity.

Step 3: Sketching the Curve

To sketch the curve, analyze typical values for \( t \): - If \( t = 0 \), \( x = 0 \), \( y = 1 \).- If \( t = 1 \), \( x = 1 \), \( y = 0 \).- If \( t = -1 \), \( x = 1 \), \( y = 0 \).- As \(|t| \to \infty\), both \( x \) and \( y \) will diverge because \( x \) increases and \( y \to 1 \).Plotting these values shows a V-like shape opening upwards, centered on the y-axis.

Step 4: Eliminate the Parameter

To find the rectangular-coordinate equation, express \( y \) solely in terms of \( x \). From \( x = |t| \), we have two cases:**Case 1:** When \( t \geq 0\), \( x = t \). - Thus, \( y = |1 - t| = |1 - x| \).**Case 2:** When \( t < 0\), \( x = -t \). - Thus, \( y = |1 + t| = |1 - x| \). In both cases, the expression \( y = |1 - x| \) holds. Therefore, the rectangular equation is \( y = |1 - x| \).

Step 5: Verify the Rectangular Equation

Lastly, verify the derived equation covers the parametric setup. For any \( x = |t| \), \( y =|1-|t|| \) results in \( y = |1-x| \), satisfying all conditions. The V-shape confirms continuity across both \( t \geq 0 \) and \( t < 0 \).

Key Concepts

Rectangular-coordinate equation

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

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.

Absolute value function

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.