Question

a parametric representation of a curve is given. $$ x=2 t, y=3 t ;-\infty

Short answer

The curve is a straight line given by \( y = \frac{3}{2}x \).

Step-by-step solution

Identify the Parametric Equations

The parametric equations are given as: \( x = 2t \) and \( y = 3t \). These equations express \( x \) and \( y \) as functions of the parameter \( t \).

Eliminate the Parameter

To find the relation between \( x \) and \( y \), solve one of the parametric equations for \( t \). Let's solve \( x = 2t \): \( t = \frac{x}{2} \). Substitute this into \( y = 3t \): \( y = 3\left(\frac{x}{2}\right) = \frac{3}{2}x \).

Write the Resulting Equation

The equation relating \( x \) and \( y \) is \( y = \frac{3}{2}x \). This is the equation of a straight line in Cartesian coordinates without any parameter.

Verify the Domain Restrictions

The parameter \( t \) can range from \(-\infty\) to \(+\infty\). This means there are no restrictions to the values of \( x \) and \( y \) as \( t \) takes all real values. Hence, the line is defined for all real \( x \).

Key concepts

Curve Sketching

Curve sketching is a useful technique in mathematics that helps visualize the path a set of equations creates. When dealing with parametric equations like the ones given, our goal is to understand what shape or curve is described in space. The parametric equations express both the horizontal (x-axis) and vertical (y-axis) coordinates as functions of a parameter, which commonly is "t". By altering the value of "t", the curve is traced out point by point in the graphical plane.

To sketch the curve, you evaluate the equations at several values of "t" and plot these points on the graph. For example, with the equations \( x = 2t \) and \( y = 3t \), you'll find points like (0,0) when \( t = 0 \), (2,3) when \( t = 1 \), and (-2,-3) when \( t = -1 \). These points should be drawn and connected in order as "t" increases, giving us a clearer picture of the curve. In this case, the curve is a straight line through these points.

Understanding the path of a curve through parametric equations is important, especially in physics and engineering, where objects follow precise trajectories.

Eliminating the Parameter

Eliminating the parameter in parametric equations is a process used to find a relationship between the dependent variables, usually x and y, without any parameter involved. This step is crucial because it converts the parametric form into a more familiar Cartesian equation.

The initial step involves solving one of the equations for the parameter ("t", in most cases). For instance, given \( x = 2t \), we rearrange it as \( t = \frac{x}{2} \). Next, substitute this expression for "t" in the other parametric equation \( y = 3t \). Doing this yields \( y = 3\left(\frac{x}{2}\right) = \frac{3}{2}x \).

After the parameter is eliminated, we arrive at the Cartesian equation \( y = \frac{3}{2}x \), a linear equation describing a straight line. This process simplifies complex curves into simpler expressions, making them easier to analyze.

Cartesian Coordinates

Cartesian coordinates provide a systematic way of describing the position of points on a plane. When we use a parameter to describe a curve, converting it to Cartesian coordinates gives a more straightforward representation.

In the example of \( x = 2t \) and \( y = 3t \), the task was to transition from parametric forms to \( y = \frac{3}{2}x \), which is its Cartesian form. This line equation is easily recognizable and tells us that the line has a slope of \( \frac{3}{2} \) and passes through the origin (0,0).

The Cartesian coordinate system uses an x-axis and a y-axis, both of which are perpendicular, to define the position of points in a two-dimensional space. Each point can be described simply by a pair of values (x, y), making this system highly intuitive and favored for graphing linear relationships. Through Cartesian coordinates, we can interpret and analyze the geometric nature of equations effectively.