$\"\"$

(a)

Graph the equation $\"\"$ on $\"\"$ .

$\"\"$

Graph the equation $\"\"$ on $\"\"$ .

$\"\"$

Graph the equation $\"\"$ on $\"\"$ .

$\"\"$

(b) Symmetry of $\"\"$ :

The equation $\"\"$ is symmetric with respect to the line $\"\"$ when the interval for $\"\"$ is $\"\"$ , where $\"\"$ is any real number.

$\"\"$

(c)

The equation $\"\"$ .

Substitute $\"\"$ .

$\"\"$

This substitution produces an equivalent equation, therefore $\"\"$ is symmetric with respect to the line $\"\"$ .

$\"\"$

(d)

It does not affect the other classic curves. The classic curves all consist of either a sine or cosine function. Therefore, to achieve a complete graph, they just need to be graphed foe all values of within their period. Extending the interval of to include additional values outside of the period will result in the graph repeating itself. Because the spiral of Archimedes does not contain a trigonometric function, additional values of $\"\"$ result in different values of $\"\"$ .

$\"\"$

(a)

Graph of $\"\"$ on $\"\"$ . Graph of $\"\"$ on $\"\"$ .

$\"\"$ $\"\"$

Graph of equation $\"\"$ on $\"\"$ .

$\"\"$

(b) The equation $\"\"$ is symmetric with respect to the line $\"\"$ when the interval for $\"\"$ is $\"\"$ , where $\"\"$ is any real number.

(c) The substitution $\"\"$ produces an equivalent equation, therefore $\"\"$ is symmetric with respect to the line $\"\"$ .

(d) It does not affect the other classic curves. The classic curves all consist of either a sine or cosine function. Therefore, to achieve a complete graph, they just need to be graphed for all values of $\"\"$ within their period. Extending the interval of $\"\"$ to include additional values outside of the period will result in the graph repeating itself. Because the spiral of Archimedes does not contain a trigonometric function, additional values of $\"\"$ result in different values of $\"\"$ .