Problem 21 - 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 is \(y = 2x\). The graph is a straight line passing through the origin with an increasing slope from left to right.

Step 1: Convert to Rectangular Form

Start by using the given parametric equations \(x=t\) and \(y=2t\). From the first equation \(x=t\), substitute the value of \(t\) from first equation to second equation, which gives us a new equation in terms of \(x\): \(y = 2x\).

Step 2: Sketch the Curve

Next, the rectangular equation \(y = 2x\) could be interpreted as a straight line passing through the origin (0,0) with a slope of 2 (rise over run), which means for every two units moved vertically, one unit is moved horizontally. To graph this equation, first draw a straight line passing through the origin and inclined at an angle. Since there is no restriction on \(t\), the line extends indefinitely in both directions.

Step 3: Add Orientation

Now, add orientation to the curve. With increasing values of \(t\) (from \(-\infty\) to \(+\infty\)), both \(x\) and \(y\) increase. This corresponds to moving from left to right across the cartesian plane. Thus, place an arrow pointing to the right on the line.

Key Concepts

Rectangular Form

Parametric equations describe curves by defining both the x and y coordinates as functions of a third variable, typically denoted as \(t\), which is the parameter. In this exercise, the given parametric equations are \(x = t\) and \(y = 2t\). To express these equations in rectangular form, we need a relationship between \(x\) and \(y\) without involving the parameter t directly.

We can achieve this by solving one of the parametric equations for \(t\) and then substituting this expression in the other equation. For instance, since \(x = t\), we can substitute \(t\) with \(x\) in \(y = 2t\). This substitution yields \(y = 2x\), which is a simple linear equation in the traditional rectangular form (also known as Cartesian coordinates).

• This rectangular form equation \(y = 2x\) describes the same geometric object as the parametric equations but is often easier to analyze and visualize.
• The rectangular form allows us to see the entire curve and its behavior in a single glance, without considering the parameter \(t\).

Plane Curve

A plane curve is a continuous set of points in the x-y plane that are defined by equations, such as our converted equation \(y = 2x\). Instead of focusing on the parameter \(t\), we can focus on how the x and y values interact across a two-dimensional surface.

The curve defined by \(y = 2x\) is a straight line because it can be viewed as a linear equation with no bending or curvature. This line passes through the origin (0,0) and continues infinitely in both directions, creating a path that holds the same properties everywhere on the coordinate plane.

• The slope of the line is 2, meaning for each unit increase in \(x\), \(y\) increases by two units, confirming it's a straight incline through the plane.
• Since both \(x\) and \(y\) can have any real value, the line stretches endlessly along the path dictated by the linear relationship.

Orientation of Curve

Orientation gives meaning to a curve by specifying the direction in which it is traced as the parameter \(t\) increases. It's like giving a traffic direction to a road drawn on the plane.

For a line defined by parametric equations, such as \(x = t\) and \(y = 2t\), orientation indicates how the curve is "drawn" as \(t\) progresses. When \(t\) increases from \(-\infty\) to \(\infty\), it implies increasing values for both \(x\) and \(y\).

• In our case, as \(t\) goes from less negative towards positive infinity, the line traverses from the bottom left to the top right of the coordinate plane.
• This is normally depicted with an arrow pointing in the positive direction along the x-axis, indicating that the motion proceeds from left to right.

This orientation is crucial for understanding how the path is followed over time, giving life and direction to the abstract concept of a curve.