Question

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

Accepted Answer

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 $$x = 2 	an t. $$Substitute into the original equation: $$y = (2 	an t$$)^{2}$$ + 4 = 4 	an$$^{2}$$ t + 4. So $$the parametric equations become $$x = 2 	an t$$, $$y = 4 	an$$^{2}$$ t + 4$$, where $$t is in $$the domain where $$	an t is $$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.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rectangular and Parametric Equations

Parametrization Techniques

Quadratic Functions and Their Graphs