Beginning Activity Beginning Activity 1: The Number of Diagonals of a Polygon
A polygon is a closed plane figure formed by the joining of three or more straight lines. For example, a triangle is a polygon that has three sides; a quadrilateral is a polygon that has four sides and includes squares, rectangles, and parallelograms; a pentagon is a polygon that has five sides; and an octagon is a polygon that has eight sides. A regular polygon is one that has equal-length sides and congruent interior angles.
A diagonal of a polygon is a line segment that connects two nonadjacent vertices of the polygon. In this activity, we will assume that all polygons are convex polygons so that, except for the vertices, each diagonal lies inside the polygon. For example, a triangle (3-sided polygon) has no diagonals and a rectangle has two diagonals.
1.
How many diagonals does any quadrilateral (4-sided polygon) have?
2.
Let \(D = \mathbb{N} - \left\{ {1, 2} \right\}\text{.}\) Define \(d\x D \to \mathbb{N} \cup \left\{ 0 \right\}\) so that \(d( n )\) is the number of diagonals of a convex polygon with \(n\) sides. Determine the values of \(d(3)\text{,}\) \(d(4)\text{,}\) \(d(5)\text{,}\) \(d(6)\text{,}\) \(d(7)\text{,}\) and \(d(8)\text{.}\) Arrange the results in the form of a table of values for the function \(d\text{.}\)
3.
Let \(f\x \mathbb{R} \to \mathbb{R}\) be defined by
Determine the values of \(f(0)\text{,}\) \(f(1)\text{,}\) \(f(2)\text{,}\) \(f(3)\text{,}\) \(f(4)\text{,}\) \(f(5)\text{,}\) \(f(6)\text{,}\) \(f(7)\text{,}\) \(f(8)\text{,}\) and \(f(9)\text{.}\) Arrange the results in the form of a table of values for the function \(f\text{.}\)
4.
Compare the functions in Exercise 2 and Exercise 3. What are the similarities between the two functions and what are the differences? Should these two functions be considered equal functions? Explain.