A library of functions: Functions and their graphs

These activities were taken from the review chapter in the Calculus book that is the successor of this course's Precalculus book. In addition to the kinds of functions explored in Chapter 9: Polynomial and Rational Functions, a review of functions explored earlier in the course (e.g., sinusoidal functions) are also explored.

Power, polynomial and rational functions

This week's handout reviews the fundamental aspects of these three types of function, and polynomial and rational functions are seen as combinations of other functions.

Compositions, Inverses, and Translations of Functions

If the course had proceeded from a function-based approach, the concepts in Chapter 8 would have appeared in Chapter 1. That would have been helpful to students so that they could see patterns across the ways we act on functions. Although that's not the case, students have seen some of the techniques covered in this chapter in the earlier chapters. Perhaps this is an attempt to "draw things together."

Download the problem set here.

Applying a few of the trig identities

These 3 problems are interesting demonstrations of the usefulness of trig identities. Even better, the paths to the solutions are interesting and the solutions themselves are surprising.

Have fun.

Week 9: Using trig identities to solve trig equations

This is the week before test 2, so the session could be about preparation in terms of getting homework and other questions answered.

We'll also use trig functions and their inverses and their domains and ranges, along with trig identities, to solve trig equations.

This handout provides students with the trig identities they'll need to know, along with 10 practice problems. We'll solve these problems in ways that draw on the relevant concepts in order for a richer, broader exam prep experience.

Week 8: Sinusoidal functions (cont.) plus...

Meetings were cancelled for three of the five groups last week, so we'll complete the activities with groups who haven't done so.

Then we'll try this problem. It's about developing a sinusoidal function to match a given graph. The answer is in the first comment below.

There were also 2 problems from the text that we didn't get to from Week 7, which should be suitable for some good, productive groupwork. Let's see how those work out.

Expect a 10-minute warm-up of questions to be solved using the unit circle.

Week 7: Exploring sinusoidal functions

Well, we didn't spend enough time on the activity in week 6. There was homework to do, and my plans for networking and groupworking didn't come to pass.

This week, we'll explore transformations of the sine and cosine functions to parallel section 6.5 in the book and revisit transformations of functions in general. At the same time, we can hit those big words we didn't quite get to last week. I'm especially thinking about: period, amplitude, midline, horizontal shift, and phase shift.

We'll develop procedures for finding these values when given sine and cosine functions in a variety of forms, and vice versa. Then we'll apply the procedures to develop models of realistic situations. Textbook problems 28 and 35 (section 6.5, p. 277) are fine candidates. Download them here.

Week 6: Exploring the trig functions

Last week, we developed our understanding of the sine and cosine functions by connecting their graphical, numerical, and geometric representations. This week we explore those functions and the trig function that composes them. We'll assemble a network-of-sorts using the graphing calculators and use that network to structure the mathematical activity using students' contributions. We move back and forth between big ideas and their instances. And we use big words like periodic, amplitude, midline, symmetry, transformation, phase shift, inverse, domain and range, global max and min, roots/zeros, and arc length.

This link takes you to a collection of trig-related applets, some of which we'll use in class to explore these functions.

Wolfram|Alpha is also a handy tool for exploring functions and other mathematical objects.

Students who haven't completed the activity from week 5 should do so before heading on to this exploration.

[Nothing to download this time.]

Week 5: Multiple representations of two trig functions

Although this may initially appear to be a rather conventional activity, we're guided by a notion of understanding as connectedness among representations. As such, we seek to represent two trig functions in three ways.

Download here.

Week 4: Exploring exponential and logarithmic functions

This exploration activity builds on the model-eliciting activity from Week 3. Now that students have developed an exponential model, they'll explore aspects of the model and its inverse, a logarithmic function.

Download the activity here: Corn Production 2

Again, graphing calculators should be made available to students as they complete the activity.

Also this week, a conversation about the use of graphing calculators (i.e., Why might graphing calculators be banned?) and conceptions of what it means to struggle (i.e., good thing or bad thing?).

Week 3: What it means to collaborate + Developing the exponential function

The Corn Production Problem is a collaborative, thought-revealing activity (in progress) about exponential functions (I hope). You can find it here.

Graphing calculators should be made available to students as they solve the problem.

Week 2: Beliefs about math & Student Development

Beliefs survey here: What beliefs about the nature of mathematics do students have as a result of their experiences in math classes? How are these in/consistent with what mathematics is really about?

Student Development map here: I'll be focusing on leveraging opportunities for agency and participation in the classroom.

Week 1: Clinical Interview

Here are the clinical interview questions: The outcomes of this activity are fruitful for all kinds of topics of discussion. We've focused primarily on knowing math as knowing how AND knowing why, or understanding as sense-making.