Problem 11 - Precalculus with Calculus Previews

Question

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=t^{3}+t+4, y=-2\left(t^{3}+t\right) $$

Step-by-Step Solution

Answer

The rectangular equation is \( y = -2x + 8 \).

Step 1: Express x in terms of t

Start with the first parametric equation given: \( x = t^{3} + t + 4 \). Isolate the terms involving \( t \) by subtracting 4 from both sides to get \( x - 4 = t^{3} + t \).

Step 2: Substitute into the equation for y

Use the expression for \( t^3 + t \) derived from the \( x \) equation. Substitute \( x - 4 \) for \( t^3 + t \) in the \( y \) equation: \( y = -2(t^{3} + t) \). This becomes \( y = -2(x - 4) \).

Step 3: Simplify the equation

Distribute the -2 on the right side: \( y = -2x + 8 \). This gives the rectangular equation in terms of \( x \) and \( y \).

Key Concepts

Eliminating Parameters

In the realm of parametric equations, we often express coordinates in terms of a parameter, usually denoted as \( t \). The "parameter" is a variable that isn't \( x \) or \( y \) in the equation but is used to describe them, enabling a pathway for more dynamic representations of curves. By eliminating the parameter, we essentially remove \( t \) from the equations.

To simplify, think of eliminating the parameter as our goal to convert two separate equations into one single equation involving only \( x \) and \( y \). This transformation is essential for linking parametric forms back to their cartesian or rectangular equation equivalent. In our exercise, we start with equations \( x = t^3 + t + 4 \) and \( y = -2(t^3 + t) \). We isolate \( t^3 + t \) from the first equation to replace it in the second. This process consolidates the information, making the connection between the variables \( x \) and \( y \) clearer.

Rectangular Equations

Switching from parametric to rectangular equations translates the information from a multi-step expression into a simpler single equation in terms of two variables, usually \( x \) and \( y \). The eventual goal is to integrate the results into a usable format like \( y = f(x) \).

In the given problem, our path to a rectangular equation was to first manipulate the parametric equation \( x = t^3 + t + 4 \). We isolated the terms involving \( t \) to express them as \( t^{3}+t = x - 4 \). From the \( y \) equation \( y = -2(t^3 + t) \), substituting the earlier result ends with \( y = -2(x-4) \), yielding the clean, rectangular equation: \( y = -2x + 8 \).

This equation is simple compared to the original parametric form, which makes it easier to use standard methods for graphing or further algebraic manipulation.

Graphing

Graphing parametric equations involves plotting points derived from both \( x = f(t) \) and \( y = g(t) \). This method often provides a detailed plot as figures evolve uniquely over the "parameter time" \( t \). Converting to the rectangular form, graphing becomes a typical linear 2D graph exercise.

For the rectangular equation \( y = -2x + 8 \) derived earlier, graphing is more straightforward. It's the equation of a straight line where:

• The slope of the line is -2, indicating the line falls at a steep angle.
• The y-intercept is 8, marking the point where the line crosses the y-axis.

By starting at the y-intercept, you can move down 2 units for every 1 unit you move right, representing how graphs of lines can be constructed efficiently from straightforward rectangular equations.

Algebra

Algebra serves as the essential toolkit in manipulating parametric equations into rectangular forms. It allows for clever restructuring and substitutions to simplify complex relationships into more recognizable formats.

Working through our problem required algebraic talents: subtracting 4 from both sides in \( x = t^3 + t + 4 \), and then using substitution for \( t^3 + t \) into the \( y \) equation. Further algebraic decisions simplify this to \( y = -2x + 8 \), an equation now stripped of parameters and readable clearly in x and y terms.

Without algebra, the leap from parametric to rectangular would be less intuitive. Mastery of algebra empowers students to not just follow a series of steps, but to understand why each step matters and how each builds upon the insights that lead to the final rectangular form.