TBIL-Cal Parametric/Vector Equations (CO1)

-values may be thought of representing the moment of time when an object is located at a particular position, and the graph may be thought of as the path the object travels throughout time.

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Activity 7.1.4.

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Earlier we obtained the same graphs for the

x y

equation

y = 2 - x^{2}

and the parametric equations

x = t - 2

and

y = - t^{2} + 4 t - 2 .

Do the following steps to find out why.

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(a)

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Which of the following equations describes

t

in terms of

x ?

$t = x - 2$
$t = x + 2$
$t = 2 x$
$t = - 2 x$

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(b)

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Which of these is the result of plugging this choice in for

t

in the parametric equation for

y ?

$y = - x + 2^{2} + 4 x + 2 - 2$
$y = - (x + 2)^{2} + 4 (x + 2) - 2$
$y = - x^{2} + 2^{2} + 4 x + 4 \cdot 2 - 2$

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(c)

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Show how to simplify this choice to obtain the equation

y = 2 - x^{2} .

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Fact 7.1.5.

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One method of graphing parametric equations

x = f (t)

and

y = g (t)

is to combine them into a single equation only involving

x

and

y,

and using your usual graphing techniques.

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Activity 7.1.6.

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Parametric equations have the advantage of describing paths that cannot be described by a function

y = h (x) .

One such example is the graph of

x = 3 \sin (π t)

and

y = - 3 \cos (π t) .

(Use technology or the approximation

\sqrt{2} \approx 0.707

to approximate coordinates as needed.)

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(a)

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Complete the folowing table.

Table 164. Chart of approximate

x

and

y

values

$t$	$x$	$y$
$0$
$1 / 4$
$1 / 2$
$3 / 4$	$2.12$	$2.12$
$1$
$5 / 4$
$3 / 2$
$7 / 4$
$2$

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(b)

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Plot these

(x, y)

points in the

x y

plane and connect the dots to draw a sketch of the graph.

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(c)

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What do you obtain by plugging the parametric equations into the expression

x^{2} + y^{2} ?

$x^{2} + y^{2} = - 6 \sin (π x) \cos (π x)$
$x^{2} + y^{2} = 9$
$x^{2} + y^{2} = 6 \sin (π x) \cos (π x)$
$x^{2} + y^{2} = 0$

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(d)

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Which of these describes the

x y

equation and graph given by these parametric equations?

a parabola
a line
a circle
a square

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(e)

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The graph of these parametric equations cannot be described by a function. Why?

The graph fails the vertical line test.
The graph fails the horizontal line test.
The graph doesn’t extend vertically to $+ \infty .$
The graph doesn’t extend horizontally to $- \infty .$

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Definition 7.1.7.

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The parametric equations

x = f (t)

and

y = g (t)

are sometimes written in the form of the vector equation

\vec{r} = ⟨ f (t), g (t) ⟩ .

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For example, the parametric equations

x = 3 \sin (π t)

and

y = - 3 \cos (π t)

may be combined into the single vector equation

\vec{r} = ⟨ 3 \sin (π t), - 3 \cos (π t) ⟩ .

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Activity 7.1.8.

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Consider the vector equation

\vec{r} = ⟨ 2 t - 3, - 6 t + 13 ⟩ .

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(a)

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What are the corresponding parametric equations?

$x = 2 t - 3$ and $y = - 6 t + 13$
$y = 2 t - 3$ and $x = - 6 t + 13$
$x y = 2 t - 3 - 6 t + 13$
Vector equations cannot be converted into parametric equations.

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(b)

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Draw a table of

t,

x,

and

y

values with

t = 0, 1, 2, 3, 4 .

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(c)

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Plot these

(x, y)

points in the plane and connect the dots to sketch the graph of this vector equation.

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(d)

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Solve for

t

in terms of

x

and plug into the

y

parametric equation to show that this is the vector equation for the line

y = - 3 x + 4 .

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Subsection 7.1.2 Videos

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Figure 165. Video for CO1

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Subsection 7.1.3 Exercises

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Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/CO1/

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