Unit 4: Conic Sections, Vectors & Matrices

In Unit 4, students explore function types that expand their understanding of the function concept. The implicitly-defined functions study of conic sections does not include all of the characteristics found in a full analytical treatment of conic sections. It is limited to quantities found in the equations involving two variables that are needed to sketch graphs of the functions.  Conic sections are limited to those whose graphs have vertical and horizontal lines of symmetry.  Another major function type in this unit involves matrices mapping a set of input vectors to output vectors. The capacity to map large quantities of vectors instantaneously is the basis for vector-based computer graphics. While students may see their favorite video game character trip and fall or seemingly move closer or farther, matrices implement a rotation on a set of vectors or a dilation on a set of vectors. The course framework limits matrices, vectors, and linear transformations to R^2 , but R^n can be explored if desired. The power of matrices to map vectors is not limited to graphics but to any system that can be expressed in terms of components of vectors such as electrical systems, network connections, and regional population distribution changes over time. Vectors and matrices are also powerful tools of data science as they can be used to model aspects of complex scientific and social science phenomena. Discussion on the basis of a vector space is bypassed but can be developed if time allows since determining if vectors are linearly independent reinforces multiple topics such as determinants and inverses. Time permitting, additional depth in this unit can be achieved by exploring parametric functions, which have multiple dependent variables’ values paired with a single input variable or parameter. Modeling scenarios with parametric functions allows students to explore change in terms of components. This component-based understanding is important not only in calculus but in all fields of the natural and social sciences where we seek to understand one aspect of a phenomenon independent of other confounding aspects.  The potential application of parametric functions is extensive but is limited to planar motion in this course.