In Exercises 53–56, find two different sets of parametric equations for each rectangular equation. y = x² + 4

Verified step by step guidance

1. Recognize that the given rectangular equation is a parabola: \(y = x^{2} + 4\). Our goal is to express both \(x\) and \(y\) in terms of a parameter \(t\) to form parametric equations.
2. For the first set of parametric equations, let the parameter \(t\) represent \(x\). So, set \(x = t\). Then, substitute \(t\) into the original equation to find \(y\): \(y = t^{2} + 4\). Thus, the first set is \(x = t\), \(y = t^{2} + 4\).
3. For the second set, choose a different parameterization. For example, let \(t\) represent \(y\). From the original equation, solve for \(x\) in terms of \(y\): \(x = \pm \sqrt{y - 4}\). To avoid ambiguity, pick one branch, say the positive root, and set \(y = t\). Then, \(x = \sqrt{t - 4}\). So the second set is \(x = \sqrt{t - 4}\), \(y = t\), with the domain \(t \geq 4\).
4. Alternatively, for the second set, you can introduce a trigonometric parameterization by letting \(x = 2 \tan t\). Substitute into the original equation: \(y = (2 \tan t)^{2} + 4 = 4 \tan^{2} t + 4\). So the parametric equations become \(x = 2 \tan t\), \(y = 4 \tan^{2} t + 4\), where \(t\) is in the domain where \(\tan t\) is defined.
5. Summarize that parametric equations are flexible and can be chosen by assigning the parameter to either \(x\), \(y\), or a function of \(t\), then expressing the other variable accordingly to satisfy the original rectangular equation.

Key Concepts

Rectangular and Parametric Equations

A rectangular equation relates x and y directly, while parametric equations express both x and y as functions of a third variable, usually t. Converting between these forms allows for different representations of the same curve, useful in analyzing motion or graphing.

Parametrization Techniques

Parametrization involves choosing a parameter t and expressing x and y in terms of t to satisfy the original equation. Common methods include letting x = t and solving for y, or using trigonometric or polynomial functions to create alternative parameter sets.

Quadratic Functions and Their Graphs

The given equation y = x² + 4 is a parabola shifted upward by 4 units. Understanding the shape and properties of quadratic functions helps in selecting appropriate parametric forms that trace the curve accurately.