Problem 55 - Algebra and Trigonometry

Question

Find two different sets of parametric equations for each rectangular equation. \(y=x^{2}+4\)

Step-by-Step Solution

Answer

The two different sets of parametric equations for the rectangular equation \(y=x^{2}+4\) are: (1) - \(x = t\) and \(y = t^{2}+4\), (2) - \(x = t-1\) and \(y = (t-1)^{2}+4\).

Step 1: Finding the first set of parametric equations

Take \(x = t\), the most general substitution. Substituting this into the given equation for \(x\), we get \(y = t^{2}+4\). Thus, one set of parametric equations is \(x = t\) and \(y = t^{2}+4\).

Step 2: Finding the second set of parametric equations

For obtaining a different set of parametric equations, consider \(x = t-1\). Substituting \(x = t-1\) into the given equation, we get \(y = (t-1)^{2}+4\). Therefore, a second set of parametric equations for the given rectangular equation is \(x = t-1\) and \(y = (t-1)^{2}+4\).

Key Concepts

Rectangular Equations

Rectangular equations, also known as Cartesian equations, are mathematical expressions that define a relationship between two variables on a Cartesian coordinate system. These equations typically involve an expression in terms of x and y, such as the equation of a line or a curve. For example, the given rectangular equation in the exercise is \[ y = x^2 + 4 \].This is a quadratic equation, which describes a parabolic curve.

• Rectangular equations are often used to describe the relationship between two variables on a graph.
• They serve as the standard form for expressing equations in two-dimensional space.

These equations allow us to analyze and visualize data in both algebraic and geometric terms, offering a more intuitive understanding of mathematical relationships.

Parametrization

Parametrization involves representing equations in terms of a third variable, typically denoted as \( t \). This allows for a different perspective on how variables relate to each other. Instead of expressing y solely as a function of x, we use another variable to define both x and y. For instance, in the examples provided:

• First parametrization: \( x = t \) and \( y = t^2 + 4 \)
• Second parametrization: \( x = t - 1 \) and \( y = (t-1)^2 + 4 \)

By introducing \( t \), parametrization provides:

• Flexibility in describing the path or motion of a curve.
• Alternatives in establishing relationships between variables, which can simplify solving problems.

Parametrization is particularly useful in physics and engineering where describing positions and trajectories are essential.

Quadratic Functions

Quadratic functions are polynomial equations of degree 2, and they form parabolic graphs. The standard form of a quadratic function is:\[ y = ax^2 + bx + c \]In the exercise, our quadratic function is simpler:\[ y = x^2 + 4 \]Here, the term \( x^2 \) defines the characteristic shape of the parabola, opening upwards.

• Quadratics always create a U-shaped graph, known as a parabola.
• The coefficient of \( x^2 \) determines the width and direction of the parabola.

Understanding quadratic functions lays the groundwork for comprehending more complex polynomials. They also frequently appear in various applications such as calculating projectiles, optimizing areas, and graphing motions.

Algebraic Substitutions

Algebraic substitutions are a powerful method for manipulating and solving equations. By substituting one expression with another equivalent expression, we can simplify problems or create new forms of equations. In the exercise above, we use substitutions to derive different sets of parametric equations.

• First substitution: By letting \( x = t \), the equation becomes \( y = t^2 + 4 \).
• Second substitution: By letting \( x = t-1 \), it transforms to \( y = (t-1)^2 + 4 \).

Substitutions are very versatile.
They help in

• Transforming complex equations into more manageable ones.
• Facilitating the conversion between different forms of equations, such as from rectangular to parametric.

Mastering algebraic substitutions not only streamlines problem-solving but also deepens understanding of algebraic concepts.