Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=t, y=-4 t$$

Step-by-Step Solution

Answer

The rectangular equation for the given parametric equations \(x = t\) and \(y = -4t\) is \(y = -4x\), and the graph is a line running from top to bottom across the graph as \(t\) increases.

Step 1: Plotting the graph

Start by selecting a range of \(t\) values and plot the corresponding \(x(t)\) and \(y(t)\) values on a graph. Here is an example: If \(t = -2\), then \(x=-2\) and \(y=8\). If \(t = 0\), then \(x=0\) and \(y=0\). If \(t = 2\), then \(x=2\) and \(y=-8\).

Step 2: Indicating orientation of curve

Orientation of the curve is determined by the direction in which the curve is drawn as \(t\) increases from negative to positive infinity. In this case, as \(t\) increases, \(x\) increases and \(y\) decreases. Thus, the orientation of the curve goes from top to bottom along the line.

Step 3: Eliminating parameter to get rectangular equation

Given the parametric equations \(x = t\) and \(y = -4t\), we can eliminate the parameter \(t\) by substitifying \(t\) in the second equation with \(x\) from first one to get the rectangular equation. Therefore, \(y = -4x\) is the rectangular equation.

Key Concepts

Graphing Parametric Equations

Graphing parametric equations is a beautiful and intriguing way to represent curves using parameters, typically denoted by a variable such as \( t \). Unlike traditional graphs where you relate \( x \) and \( y \) directly, parametric equations express both \( x \) and \( y \) as functions of this parameter. For example, given \( x = t \) and \( y = -4t \), each value of \( t \) will produce a corresponding \( x \) and \( y \) that can be plotted to form a curve.
The process involves selecting different \( t \) values within a range and plotting their results:

• Choose distinct values for \( t \) such as \( -2, 0, \) and \( 2 \).
• Calculate \( x(t) \) and \( y(t) \) for these \( t \) values.
• Plot the points \((-2, 8), (0, 0), (2, -8)\) on a graph.

By connecting these points with a smooth line, you visualize the curve formed by these parametric equations. It's important to indicate the curve's orientation too, which in this case, as \( t \) increases, moves from top to bottom. This knowledge helps in understanding both continuity and directionality of the curve.

Eliminating Parameters

Eliminating the parameter in parametric equations involves removing the \( t \) variable to find a direct relationship between \( x \) and \( y \), leading to a more familiar rectangular form. It's a handy method to simplify understanding of complex curves.
Given the equations \( x = t \) and \( y = -4t \), eliminating the parameter is done by isolating \( t \) in one equation and substituting it into the other. Since \( x = t \), you can directly replace \( t \) in the second equation:

• Express \( t \) from the first equation: \( t = x \).
• Substitute \( t \) in the second equation to get \( y = -4x \).

This conversion helps us derive the rectangular (or Cartesian) form of the represented curve. It makes graphing and understanding connections more intuitive by seeing how changes in \( x \) directly affect \( y \).

Rectangular Equations

Rectangular equations cut through the complexity of parametric forms by expressing a direct algebraic relationship between \( x \) and \( y \). In the context of the equations \( x = t \) and \( y = -4t \), converting them gives us the rectangular equation \( y = -4x \). This represents the same line on the graph as our parametric form but simplifies its interpretation by removing the parameter \( t \).
In rectangular form, it's straightforward to see the relationship. The equation \( y = -4x \) indicates a linear graph with a constant slope of \(-4\), reflecting a straight line that passes through the origin. The implications are clear: for every increase in \( x \) by 1 unit, \( y \) decreases by 4 units.
Understanding this transformation is crucial for analyzing data and solving mathematical problems, as it provides a clearer, more conventional means to deduce results effectively without the added layer of the parameter.