Problem 22 - Algebra and Trigonometry

Question

Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that \(-\infty< t <\infty.\)) \(x=t, y=-2 t\)

Step-by-Step Solution

Answer

The rectangular equation that represents the same information as the parametric equations \(x=t, y=-2 t\) is \(y=-2x\). The graph of this equation is a line with a downward slope (slope=-2), and its orientation as t increases is from right to left.

Step 1: Eliminate the Parameter

The first step is to eliminate the parameter t. To do that, recognize that the parameter equations \(x=t\) and \(y=-2t\) can be combined by substituting the value of t from the first equation into the second. Substituting \(t=x\) we get \(y=-2x\).\n

Step 2: Translate into Rectangular Equation

The second step is translating the equation into a rectangular form. The rectangular form of the equation after substituting the parameter is \(y=-2x\). This is the equation of a line with a slope of -2.\n

Step 3: Sketch the Graph

Having found the rectangular equation, it's time to sketch this line in the xy-plane. The line starts at the origin and slopes downwards to the left, indicating the slope of -2. The orientation of the line (as t increases) is from right to left.

Key Concepts

Eliminate the Parameter

When dealing with parametric equations like the given set, x=t and y=-2t, the goal often involves eliminating the 'parameter,' which in this case is t. This step is crucial for simplifying equations into a more familiar form, and it's often done by expressing the variable in terms of one parameter and then replacing that parameter in the other equation.

For instance, because x is equal to t, directly substitute the expression of x in place of t in the equation for y, resulting in a new equation: y=-2x. This elimination process yields a relationship between x and y without the parameter, providing a straightforward path to sketching the curve and understanding its behavior.

Rectangular Equation

A rectangular equation is simply an equation expressed in terms of x and y, as opposed to parameters or other variables. It's the standard form for most graphs and functions taught in algebra. In our exercise, the elimination of the parameter t leads us to the rectangular equation y=-2x.

This expression represents a line with a slope of -2, showing a direct proportionality between x and y. Understanding the nature of the rectangular equation allows us to visualize the graph more intuitively and analyze the behavior of the line, such as its slope and y-intercept.

Graphing Parametric Equations

Graphing parametric equations is different from graphing standard rectangular equations because each value of t provides a set of x and y coordinates, thus plotting points that form a curve.

To sketch our given set of parametric equations:

• Start with various values of t and calculate the corresponding x and y values.
• Plot these points on the coordinate plane.
• Connect the points smoothly, considering the relation between them, which reveals the shape and directionality of the curve.

After eliminating the parameter, as previously explained, and deriving the rectangular equation, the graph becomes a simple straight line, making plotting even more straightforward.

Orientation of a Curve

The orientation of a curve is the direction in which the curve is traversed as the parameter increases. In parametric forms, this is crucial because unlike regular functions, parametric equations can describe curves that loop back on themselves or have other complex behaviors.

In our exercise, the orientation of the line based on the values of t clarifies the way we should draw arrows on the graph. Since the equation y=-2x results from increasing values of t (to the right on the x-axis), and recognizing that y is twice as negative as x (hence a downward slope), we draw arrows pointing to the left, reflecting a decrease in the value of x and y as t increases.