Problem 71 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=1-2 x^{2}$$

Step-by-Step Solution

Answer

The set of parametric equations representing the given rectangular equation \(y=1-2x^{2}\) when \(t=x\) is \(x=t\) and \(y=1-2t^{2}\). When \(t=2-x\), the parametric equations are \(x=2-t\) and \(y=-7+8t-2t^{2}\).

Step 1: Substitution with \(t=x\)

For the first part of the exercise, the substitution \(t=x\) will be applied. In other words, wherever \(x\) appears in the rectangular equation, it will be replaced with \(t\). So, the original equation transforms to \(y=1-2 t^{2}\). Thus, the parametric equations are \(x=t\) and \(y=1-2t^{2}\).

Step 2: Substitution with \(t=2-x\)

For the second part of the exercise, the substitution \(t=2-x\) will be applied. This requires us to solve for \(x\) in terms of \(t\), from which we get \(x=2-t\). Substituting into the rectangular equation results in \(y=1-2(2-t)^{2}\). Thus, the parametric equations are \(x=2-t\) and \(y=1-2(2-t)^{2}\).

Step 3: Simplifying the Second Set of Parametric Equations

The second set of parametric equations can be simplified further. For the equation \(y=1-2(2-t)^{2}\), we can apply the binomial theorem to expand out the terms. After simplification, we find \(y=1-2(4-4t+t^{2})=1-8+8t-2t^{2} = -7+8t-2t^{2}\). So, the simplified parametric equations are \(x=2-t\) and \(y=-7+8t-2t^{2}\).

Key Concepts

Rectangular Equation

A rectangular equation is a type of mathematical expression used to represent relationships between variables in a coordinate plane using standard Cartesian coordinates, such as \(x\) and \(y\). In our exercise, we start with the rectangular equation \(y=1-2x^2\). This equation defines a parabola opening downwards, showcasing the squared term's impact on the curve's shape.
Converting rectangular equations into parametric equations makes it easier to describe the path of an object or allow for greater ease in plotting certain curves, especially when involving motion or time as parameters. Remember, a rectangular equation is the traditional way to express curves without relying on additional parameters.

Substitution

Substitution is a powerful algebraic technique that involves replacing a variable in an equation with another expression to simplify or solve it. In this exercise, we're converting a rectangular equation into parametric form by substituting for \(x\).
First, we use the substitution \(t=x\), allowing us to directly replace \(x\) with \(t\) in our rectangular equation. This gives us a parametric form where \(x=t\) and \(y=1-2t^2\).

• Substitution \(t=x\): \(x=t\), \(y=1-2t^2\)

In another approach, we use \(t=2-x\), which might seem slightly more complex. Here, to make the substitution, we rearrange to express \(x\) in terms of \(t\), yielding \(x=2-t\). This then transforms the equation accordingly.

• Substitution \(t=2-x\): \(x=2-t\), inserting into \(y\) gives a more complex expression.

This method helps in cases where altering the variable simplifies the equation or reveals other relationships inherent in the original equation.

Binomial Theorem

The binomial theorem is a key mathematical formula used to expand expressions raised to a power, specifically binomials of the form \((a + b)^n\). It helps simplify the process of dealing with exponents, especially higher ones. In our problem, we used the binomial theorem to expand \((2-t)^2\).
The expression \((2-t)^2\) is expanded as:

• First term squared: \(2^2 = 4\)
• Two times the product of both terms: \(2 \cdot 2 \cdot -t = -4t\)
• Second term squared: \((-t)^2 = t^2\)

This expansion is crucial when working with parametric equations to simplify them for easier interpretation or computation. Using the binomial theorem reduces the complexity of polynomial expressions and ensures accuracy in calculations.

Expansion and Simplification

Expanding and simplifying an equation involves breaking down complex expressions into simpler forms and combining like terms. It's a significant step in solving or reducing equations to an understandable size.
In our task, after applying the substitution \(t=2-x\) and using the binomial theorem, we end up with the expression \(y=1-2(4-4t+t^2)\). We then follow these steps:

• Distribute \(-2\) across the terms inside the parentheses: \(-2 \cdot 4, -2 \cdot -4t, -2 \cdot t^2\)
• Simplify: This results in \(y=1-8+8t-2t^2 = -7+8t-2t^2\)

Through these steps, the parametric equation is reduced to a clear and concise format: \(y=-7+8t-2t^2\).
Simplification is valuable as it often reveals patterns or solutions that might not be immediately clear in more complicated expressions.