Learning Resources
Overview
To date, you have studied a few different functions; namely, linear, quadratic, exponential and absolute value. This chapter of study will serve to build on the knowledge you have gained and extend it to include any polynomial function.
It might be useful to write down the tools you have discovered for working with various types of functions. This list should include the transformational skills used to help you work with quadratic, absolute-value, exponential or trigonometric functions.Brainstorm with others in your class to develop a complete list of strategies. In this unit, new tools for working with functions will be created. As each one is discovered, it should be added, along with an example of how the tool is used..
This unit has six sections as outlined below.
Section 2.1: explores combinations of functions by adding, subtracting and multiplying functions. Quadratic functions are revisited, extending and building on the work already completed. Also discussed, are the shapes of cubic and quartic polynomials.
Section 2.2: deals with methods for solving higher-order polynomial equations. These solution methods are then extended to finding the solution to quadratic and other polynomial inequalities.
Section 2.3: introduces the composition of functions. You will investigate composite functions using equations, tables and graphs. These activities will enable you to visualize the composition of functions.
Section 2.4: investigates instantaneous rates of change from the perspective of a limit. This section builds on the concepts you have explored in Mathematical Modeling Book 3. At the end of this section you will have an intuitive and visual understanding of what is meant by a limit.
Section 2.5: deals with instantaneous rates of change, the Power Rule, and the equation and graph of the slope function. The term derivative is introduced, as you find expressions for the derivatives of various polynomial functions. You will also compare the graph of a function with the graph of the slope function of that function. Intercepts, slopes, and maximum and minimum values are examined to aid in this comparison.
Section 2.6: examines graphs of polynomial functions with the aid of the Power Rule and the slopes of tangents. You will discuss domain and range, intercepts, critical points, and intervals of increase and decrease. These results will be used to sketch graphs of polynomial functions and make generalizations about their shapes.
Although many of these topics may appear confusing to you now, they will be developed through a visual approach in many of the Investigations. As well, graphing technology will be a great benefit here as it can simplify the completion of much of the work.
Graph paper is essential for many activities in this unit. Be sure to have some on hand each day.
This unit of study should take about 20 - 25 hours to complete.