Problem 72 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=2-5 x^{2}$$

Step-by-Step Solution

Answer

The set of parametric equations when \(t = x\) is \[x = t\], \[y = 2 - 5 t^{2}\]. And when \(t = 2 - x\), the set of parametric equations is \[x = 2 - t\], \[y = 2 - 5 (2 - t)^{2}\].

Step 1: Substitute \(t = x\) in the Rectangular Equation

The given rectangular equation is \(y = 2 - 5 x^{2}\). Substituting \(t = x\) in the rectangular equation, this gives the new equation: \[y = 2 - 5 t^{2}\]. Thus, the set of parametric equations is: \[x = t\], \[y = 2 - 5 t^{2}\].

Step 2: Substitute \(t=2-x\) in the Rectangular Equation

Now, for \(t = 2 - x\), rearrange to get \(x = 2 - t\). Substitute this back into your given rectangular equation to get: \(y = 2 - 5 (2 - t)^{2}\). Thus, the set of parametric equations is: \[x = 2 - t\], \[y = 2 - 5 (2 - t)^{2}\].

Key Concepts

Rectangular Equation

A rectangular equation, commonly referred to in precalculus as a Cartesian equation, is a mathematical expression that involves x and y coordinates on the Cartesian plane. For instance, the equation \(y = 2 - 5x^2\) is a rectangular equation because it expresses y directly in terms of x.

Rectangular equations are fundamental in understanding the relationship between two variables and often represent curves or lines on a graph. The simplicity of these equations makes them appealing for direct evaluation and graphing, allowing us to visualize the interplay between the variables. By solving for one variable in terms of the other, we can easily describe the trajectory or shape of the graph that the equation represents.

Precalculus

Precalculus serves as the foundation for calculus, consisting of mathematical concepts and skills that prepare students for the study of limits, derivatives, and integrals. The subject encompasses a variety of topics including functions, algebraic structures, trigonometry, sequences, and series.

In the context of our example, precalculus includes the manipulation and conversion of equations from one form to another, such as turning a rectangular equation into parametric equations. Understanding these conversions is crucial for students as they provide alternative ways of representing mathematical relationships, leading to different methods of analyzing and solving problems, especially in calculus.

Substitution Method

The substitution method is a powerful algebraic tool often used in calculus and precalculus for solving systems of equations, converting equation forms, or integrating and differentiating expressions. The method involves replacing one variable with another or expressing one variable in terms of another to facilitate the simplification and solving of an equation.

In the given example, we employed the substitution method by setting \(t\) equal to \(x\) and \(2-x\), which are different parametrizations of the rectangular equation \(y = 2 - 5x^2\). This method allowed us to convert the original rectangular equation into sets of parametric equations. Through substitution, solving for y in terms of t becomes a straightforward process, and we can represent the same relationship between x and y using a parameter t.

Graphing Equations

Graphing is a crucial skill in precalculus that involves plotting equations onto the coordinate plane to visualize their solutions and the relationships between variables. Different forms of equations, such as rectangular or parametric, can be graphed to reveal various characteristics of the function they represent.

The graph of a rectangular equation like \(y = 2 - 5x^2\) showcases a parabola, opening downwards due to the negative coefficient in front of \(x^2\). On the other hand, parametric equations introduce a parameter, typically denoted by t, which describes the x and y coordinates in terms of a single variable. Graphing these equations requires evaluating both x(t) and y(t) over a range of t values – a process that can unveil more complex curves and trajectories on the plane that are not as immediately evident from the rectangular form alone.