Consider an astroid with parametric equations $$x=3\cos^3(t),\;\;y=3\sin^3(t).$$

(a) Find the slope of the tangent line to the astroid in terms of $t$.

Slope =

(b) List all of the points $(x, y)$ where the tangent line is horizontal. In entering your answer, list the points starting with the smallest value of $x$. If two or more points share the same value of $x$, list those points starting with the smallest value of $y$. If any blanks are unused, type an upper-case "N" in them.

Point 1: $(x, y) =$ (,)

Point 2: $(x, y) =$ (,)

Point 3: $(x, y) =$ (,)

(c) List all of the points $(x, y)$ where the tangent line is vertical. In entering your answer, list the points starting with the smallest value of $x$. If two or more points share the same value of $x$, list those points starting with the smallest value of $y$. If any blanks are unused, type an upper-case "N" in them.

Point 1: $(x, y) =$ (,)

Point 2: $(x, y) =$ (,)

Point 3: $(x, y) =$ (,)

(d) List all of the points $(x, y)$ where the slope of the tangent line is 1. In entering your answer, list the points starting with the smallest value of $x$. If two or more points share the same value of $x$, list those points starting with the smallest value of $y$. If any blanks are unused, type an upper-case "N" in them.

Point 1: $(x, y) =$ (,)

Point 2: $(x, y) =$ (,)

Point 3: $(x, y) =$ (,)

(e) List all of the points $(x, y)$ where the slope of the tangent line is -1. In entering your answer, list the points starting with the smallest value of $x$. If two or more points share the same value of $x$, list those points starting with the smallest value of $y$. If any blanks are unused, type an upper-case "N" in them.

Point 1: $(x, y) =$ (,)

Point 2: $(x, y) =$ (,)

Point 3: $(x, y) =$ (,)