Q.15 Explain why the parametrization ... [FREE SOLUTION]

Chapter 9: Q.15 (page 721)

Explain why the parametrization $x = \sin t + 1, y = \sin^{2} t - 4$ for $t 鈭� (- 鈭�, 鈭�)$ 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 $x = \sin t + 1, y = \sin^{2} t - 4, t 鈭� (- 鈭�, 鈭�) .$

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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91影视

Explain why the parametrization $x = \sin t + 1, y = \sin^{2} t - 4$ for $t 鈭� (- 鈭�, 鈭�)$ 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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91影视

Explain why the parametrization x=sint+1,y=sin2t-4for t鈭�(-鈭�,鈭�)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

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Statistics

Calculus

Geometry

Study anywhere. Anytime. Across all devices.

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