Students will learn how to determine the end behavior of univariate polynomials.

Intro to end behavior of polynomials by Khan Academy

End behavior of polynomials by Khan Academy

Conclude by giving your students these challenges:

In the figure below, \(C\) is the center of the circle, \(\overline{AE} = 1,\) and \(\overline{EF} = 2.\) What's the length of \(\overline{EB}?\)

Here's the solution: First, notice that \(\overline{AC} = 3.\) Next, find that \(\overline{AD} = \sqrt{3}\) using the Pythagorean theorem. Notice that \(\triangle ACD\) is a 30-60-90 triangle, with \(\angle DCA = 60^\circ.\) Also, notice that \(ECD\) is an isosceles triangle, because two of its sides are radii of the circle. We already know one of its angles, so the other two can be determined. It happens that \(\triangle ECD\) is equilateral. Thus, \(\overline{EB}\) is half the radius of the circle, so \(\overline{EB} = 1/2.\)

Suppose we're in a world with natural numbers only. Because we don't have negative numbers, an operation like x - y isn't defined for all possible choices of x and y. For example, 0 - 1 has no sensible value. But subtraction is such a useful operation, we might as well have an operation that behaves something like integer subtraction. The following is such an operation. It's often referred to as "proper subtraction." $$x \mathop{\unicode{8760}} y = \begin{cases} 0 & \text{if }x \lt y \\ x - y & \text{otherwise} \end{cases}$$

That takes care of subtraction, but what about division? We don't have rationals, so 1/2 doesn't make any sense. To handle division, we're going to define what's often called "integer division." Our definition appears below. $$\text{div}(x, y) = \left\lfloor \dfrac{x}{y} \right\rfloor$$

How can we express \(\text{max}\) and \(\min,\) in terms of only \(+,\) \(\unicode{8760},\) \(\times,\) \(\text{div}?\)