## Two Investigations of Cubic Functions

#### Activity Overview

In this activity, two interesting features of cubic functions which have three real roots are explored, namely that:

- the root of the equation of the tangent line to a cubic function at the average of two of the function's three roots turns out to be the function's third root, and
- the midpoint between the relative minimum and relative maximum points of a cubic function turns out to be the function's inflection point.

#### Before the Activity

Teachers may need to help students review methods of finding roots of a polynomial function, the quadratic formula (optional), and the midpoint formula.

Students should also know how to find the first and second derivatives of a function and how to use these derivatives to find slopes/equations of tangents to the function as well as relative minimum, relative maximum, and inflection points of the function.

#### During the Activity

Two investigations offer opportunities for students to apply their knowledge of derivatives of a function to interesting properties of cubic functions which have three real roots, namely that:

- the root of the equation of the tangent line to a cubic function at the average of two of the function's three roots turns out to be the function's third root, and
- the midpoint between the relative minimum and relative maximum points of a cubic function turns out to be the function's inflection point.

For each of (i) and (ii), an investigation starts with a specific function, f(x) = x^3 - 3x^2 - 10x + 24, and then moves to the more general case, g(x)=(x-a)*(x-b)*(x-c). CAS capabilities allow for proofs of the above features to be explored in the more general case.

This activity could be teacher-led, or could be used as a self-guided discovery for individual or small groups of students. Teachers might want to go through the first part of each investigation and have students try other parts. Or, teachers might want to use each entire investigation based on f(x) = x^3 - 3x^2 - 10x + 24 as an example and have students try other examples using other cubic functions. The function f(x) was created by expanding (x+3)(x-2)(x-4). Teachers can easily create other "nice" examples by expanding (x-a)(x-b)(x-c) where a, b, and c are integers.

#### After the Activity

Students could make conjectures and/or read about other properties of cubic functions and test ideas with TI-Nspire CAS.

Students could investigate whether similar properties hold for other types of polynomial functions. For example, if "cubic" is changed to "quartic" in the above explorations, is there any relationship(s) between the roots of a quartic g(x)=(x-a)(x-b)(x-c)(x-d) and roots of tangent lines to the quartic? Is there any relationship(s) between relative minimum, relative maximum, and inflection points?