Multiple Choice
Write parametric equations for the rectangular equation below.
10. Parametric Equations
Writing Parametric Equations
1:27 minutesProblem 5.5.55
Textbook Question
In Exercises 53–56, find two different sets of parametric equations for each rectangular equation. y = x² + 4
Verified step by step guidance1
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.
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 \(x = t\), \(y = t^{2} + 4\).
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 \(t \geq 4\).
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.
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.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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