d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

While stacked decks and two-headed coins make magic tricks easy, no gimmick can reproduce the sleight of hand that comes with hard work and practice—the true backbone of magic. Any good magician knows that. Just the same, no trick to graph a function is as effective as plotting points, especially when it comes to rational functions.

Rational functions are like polynomials with more freedom. We can have denominators, but that also means we have asymptotes, or boundaries that the function gets closer and closer to, but never actually reaches. Vertical asymptotes exist when the denominator of the function is 0. So, when given a function like

we'll know that there's a vertical asymptote when x = 1. If the highest-order x terms in the numerator and denominator have the same exponent, then there will also be a horizontal asymptote when their coefficients are divided. For instance, since both 4x + 7 and x – 1 have x¹ as their highest order x term, the horizontal asymptote will be at y = ⁴⁄₁ or y = 4. See what we mean?

Just because these functions are rational doesn't mean they're predictable, necessarily. After finding the asymptotes, students should start plotting points and hoping for the best. (Definitely not something they should do with magic, especially when fire or knives are involved.)