Four Homework Problems Over Big Ideas (CoRD)

Narges Dehdashti, Rose State College
Mike Fulkerson, University of Central Oklahoma
Cecil Phibbs, Northern Oklahoma College
Ashley Tomson, Rose State College

In this CoRD for a College Algebra course, instructors are provided with four activities in which students explore, synthesize, and make connections of key ideas in College Algebra. In Activity 1, students analyze two different applications and apply key terminology about functions. In Activity 2, students explore function transformations through a Desmos activity. In Activity 3, students connect the important characteristics of polynomials in standard and factored form with graphs and words. In Activity 4, students consider logarithmic and exponential functions to support the ideas of these functions as well as their importance as inverses of each other.

Key Characteristics of Functions

Student Version

Instructor Version

This activity focuses on characteristics of graphs such as determining if a graph is a function, domain and range, function evaluation, intercepts, intervals of increase/decrease/constant, and piecewise functions. The two provided scenarios/activities are applications of piecewise functions, but problems with a low entry point. This allows students to tie together the ideas covered in College Algebra with functions and their graphs.

Transformations of Functions

Student Version

Instructor Version

In this activity, students will use Desmos to investigate transformations of functions (vertical and horizontal translations, vertical stretches and compressions, and reflections). The goal is for students to not merely understand how to apply the various transformations, but to understand why those transformations work the way they do. The activity uses sliders in a way that allows students to see the effects of transformations quickly, while also using questions that lead students to discover the reasons the changes to the graphs occur. The activity also uses piecewise functions in several places, which is a bit unusual for an activity on transformations. The rationale for the use of piecewise functions is that many of the basic functions have various properties (symmetry, etc.) that make it difficult to determine which transformation is involved. For example, the function $f(x)=3\\sqrt x$ can be thought of as a vertical stretch (by a factor of 3) or as a horizontal compression (by a factor of 9) because $f(x)=\\sqrt{9x}$ . The same thing occurs for the basic functions $x^2$ , $|x|$ , $x^3$ , $1/x$ , etc. Similarly, several of the basic functions are either even (such as $x^2$ and $|x|$ ) or odd (such as $x^3$ and $1/x$ ) which can make reflections across the $x$ – or y-axis problematic for student understanding. For example, for the function $f(x)=x^3$ , a reflection across the $x$ -axis is the same as a reflection across the y-axis. For this reason, we have used piecewise functions to more clearly demonstrate certain transformations (namely reflections and stretches/compressions).

Polynomial Functions

Student Version

Instructor Version

In this activity, students will analyze polynomial functions. They will describe key characteristics of polynomial functions and how they relate to their graphs and will then apply those key characteristics to sketch the graph of a specific polynomial. Finally, students will work backwards from the graph of a polynomial to its equation.

Exponential & Logarithmic Functions

Student Version

Instructor Version

Students will explore exponential and logarithmic functions. Through working on tasks that require completing function tables, graphing functions, and analyzing intercepts, domains, ranges, and asymptotes, students will gain a deeper understanding of these functions. This includes exploring a practical application of these concepts through Richter scale problems, where they will use logarithmic functions to compare given earthquake magnitudes and determine the magnitudes of earthquakes based on their strength. This activity reinforces the theoretical aspects of exponential and logarithmic functions (including that they are inverses of each other) while also emphasizing their practical applications in real-world earthquake scenarios.

This work is licensed under CC BY-NC-SA 4.0