$\"\"$

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(a)

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The parametric equations are $\"\"$ and $\"\"$ .

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Find the rectangular equation.

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Consider $\"\"$ .

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$\"\"$

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Squaring on each side.

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$\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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Squaring on each side.

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$\"\"$ .

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Trigonometric identity : $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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The rectangular equation is $\"\"$ .

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Graph:

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Graph the parametric equations $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph :

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The domain of the graph is $\"\"$ .

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It is circle with Anticlockwise orientation and curve is smooth and continuous on the interval.

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The range is $\"\"$ .

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$\"\"$

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(b)

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The parametric equations are $\"\"$ and $\"\"$ .

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Graph:

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Graph the parametric equations $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph :

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The domain of the graph is $\"\"$ .

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It is semicircle with clockwise orientation and curve is smooth and continuous on the interval.

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The range is $\"\"$ .

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$\"\"$

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(c)

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The parametric equations are $\"\"$ and $\"\"$ .

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Graph:

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Graph the parametric equations $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph :

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The domain of the graph is $\"\"$ .

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It is quater circle with clockwise orientation and curve is smooth and continuous on the interval.

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The range is $\"\"$ .

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$\"\"$

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(d)

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The parametric equations are $\"\"$ and $\"\"$ .

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Graph:

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Graph the parametric equations $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph :

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The domain of the graph is $\"\"$ .

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It is semi circle with clockwise orientation and curve is smooth and continuous on the interval.

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The range is $\"\"$ .

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Observe the set of graphs of parametric equations:

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The parametric equations from (a)-(d) are similar and their rectangular equation is $\"\"$ .

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They differ in orientation and some are domain restricted.

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$\"\"$

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The parametric equations from (a)-(d) are similar and their rectangular equation is $\"\"$ .

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They differ in orientation and some are domain restricted.

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(a)

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The domain of the graph is $\"\"$ .

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It is semicircle with clockwise orientation.

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The range is $\"\"$ .

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(b)

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The domain of the graph is $\"\"$ .

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It is semicircle with clockwise orientation.

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The range is $\"\"$ .

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(c)

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The domain of the graph is $\"\"$ .

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It is quater circle with clockwise orientation.

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The range is $\"\"$ .

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(d)

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The domain of the graph is $\"\"$ .

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It is semi circle with clockwise orientation.

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The range is $\"\"$ .