Problem 3 - Precalculus Mathematics for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 t, \quad y=t+6$$

Step-by-Step Solution

Answer

(a) The curve is a straight line. (b) The rectangular equation is \( y = \frac{x}{2} + 6 \).

Step 1: Understand Parametric Equations

In the given problem, you have two parametric equations: \( x = 2t \) and \( y = t + 6 \). These equations define the coordinates \( (x, y) \) in terms of the parameter \( t \).

Step 2: Find the Relationship Between x and t

To eliminate the parameter \( t \) and relate \( x \) to \( y \) directly, first express \( t \) in terms of \( x \). Given \( x = 2t \), we can rearrange this to find \( t = \frac{x}{2} \).

Step 3: Substitute t in the Equation for y

Now that \( t \) is expressed as \( \frac{x}{2} \), substitute it into the equation for \( y \): \( y = t + 6 \). This gives you \( y = \frac{x}{2} + 6 \).

Step 4: Simplify to Find the Rectangular Equation

The rectangular equation is obtained in terms of \( x \) and \( y \). From \( y = \frac{x}{2} + 6 \), simplify to confirm that the rectangular equation is \( y = \frac{x}{2} + 6 \), which is already simplified.

Step 5: Sketch the Curve

Recognize that the equation \( y = \frac{x}{2} + 6 \) is the equation of a straight line with a slope of \( \frac{1}{2} \) and a y-intercept at 6. Plot the line on the coordinate plane, starting at (0,6) and rising 1 unit for every 2 units it runs to the right.

Key Concepts

Rectangular-Coordinate Equation

A rectangular-coordinate equation is a relationship between the Cartesian coordinates, usually expressed as \(x\) and \(y\), without involving a parameter like \(t\) in the equations. When working with parametric equations, the goal is often to convert these equations to a form where the dependence on the parameter is removed. In the problem, we start with parametric equations \(x = 2t \) and \( y = t + 6 \).

To transition to a rectangular form, you'll need to express one variable in terms of the other. Hence, you manipulate the equations to eliminate the parameter. This process forms the basis for converting parametric representations into more familiar Cartesian formats, which can be easier to analyze and graph.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations given, in terms of \(x\) and \(y\) only, and getting rid of the parameter \(t\). This usually requires a few algebraic manipulations.

In the exercise, we start with:

• \(x = 2t\)
• \(y = t + 6\)

First, we solve for the parameter \(t\) from the first equation: \(t = \frac{x}{2}\).

By substituting this expression for \(t\) in the second equation, we get \(y\) solely in terms of \(x\): \(y = \frac{x}{2} + 6\). This results in a rectangular-coordinate equation that originates from the parametric equations but has eliminated the parameter.

Straight Line Equation

A straight line equation in its simplest form is expressed as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This form shows us how the value of \(y\) changes in relation to \(x\).

In the given exercise, the resulting rectangular equation \(y = \frac{x}{2} + 6\) is already in this format, identifying it as a straight line. Here, the slope is \(\frac{1}{2}\) which means for every increase of 2 units in \(x\), \(y\) increases by 1 unit. The y-intercept is 6, which gives you the point (0,6) on the graph. Understanding these components is key to interpreting and sketching the graph correctly.

Plotting Parametric Curves

Plotting parametric curves involves creating a graph using points derived from parametric equations. These plots give a visual representation of the path traced by the equations based on varying the parameter \(t\).

Begin by plotting the rectangular-coordinate equation, \(y = \frac{x}{2} + 6\). Remember:

• The y-intercept is where the line crosses the y-axis, at (0,6).
• The slope indicates that the line rises 1 unit for every 2 units it moves to the right.

Choose a range of \(x\) values to sketch points accurately. For each chosen \(x\), calculate \(y\), plot corresponding points, and then connect them with a straight line. The resulting graph provides a clear visual understanding of how the parametric equations transform into a linear path on the Cartesian plane.