Chapter 9: Q.15 (page 721)

Explain why the parametrization $x = \sin t + 1, y = \sin^{2} t - 4$ for $t \in (- \infty, \infty)$ repeatedly traces the same small portion of the graph of the function $y = x^{2} - 2 x - 3$ .

Short Answer

Expert verified

The parameterization of the curves $\sin t = x - 1, y = \sin^{2} t - 4$ traces exactly the same curve $y = x^{2} - 2 x - 3$ as that of the equations $x = 2 t + 1, y = 4 t^{2} - 4 .$

Step by step solution

Step: 1 Given information

Consider the parametric curves $x = \sin t + 1, y = \sin^{2} t - 4, t \in (- \infty, \infty) .$

Calculation

The goal is to figure out how to parametrize the curves.

Remove the parameter $t$ from the parametric equations to parameterize the curves.

Take $x = \sin t + 1$ .

Add $- 1$ to both sides of the equation.

$x - 1 = \sin t + 1 - 1 x - 1 = \sin t + 1 - 1 x - 1 = \sin t$

Now substitute $\sin t = x - 1$ in the equation $y = \sin^{2} t - 4$ .

Then,

$y = (x - 1)^{2} - 4 y = x^{2} - 2 x + 1 - 4 y = x^{2} - 2 x - 3$

Further simplification

Here the $\sin t$ function is periodic and lies in between $- 1, 1$ . And x, y are also bounded.

Here the values that satisfy the equation $y = x^{2} - 2 x - 3$ are

Thus, the parameterization of the curves $\sin t = x - 1, y = \sin^{2} t - 4$ traces exactly the same curve $y = x^{2} - 2 x - 3$ as that of the equations $x = 2 t + 1, y = 4 t^{2} - 4$ .

Hence the explanation.

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Explain why the parametrization $x = \sin t + 1, y = \sin^{2} t - 4$ for $t \in (- \infty, \infty)$ repeatedly traces the same small portion of the graph of the function $y = x^{2} - 2 x - 3$ .

Short Answer

Step by step solution

Step: 1 Given information

Calculation

Further simplification

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Explain why the parametrization x=sint+1,y=sin2t-4for t∈(-∞,∞)repeatedly traces the same small portion of the graph of the function y=x2-2x-3.

Short Answer

Step by step solution

Step: 1 Given information

Calculation

Further simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Study anywhere. Anytime. Across all devices.

Explain why the parametrization $x = \sin t + 1, y = \sin^{2} t - 4$ for $t \in (- \infty, \infty)$ repeatedly traces the same small portion of the graph of the function $y = x^{2} - 2 x - 3$ .