A Hodgepodge of Ideas?

When I started teaching Precalculus, I thought it was no wonder the students were confused. It felt as if we jumped from topic to topic – complex numbers, trigonometry, logarithms, polar coordinates, etc. – and spent only a few days on some of them. It was just last year that I was finally able to really help my students see the connections.

For most of the course, we are studying the functions. My students will need to know them all pretty well for calculus. And the functions share many characteristics: they have a domain and a range and a graph and can be shifted right or left, up or down, or reflected in the x– or the y-axis (although a reflection in the y-axis may not return another function.) The notation for these shifts and reflections is exactly the same, no matter what the function:

y – k = f(x – h) translates h units horizontally and k units vertically

y = – f(x) reflects the function in the y-axis and yf(- x) reflects the function in the x-axis

y = cf(x) stretches or shrinks the function vertically and y = f(cx) stretches or shrinks the function horizontally

Students need so see how these are used with a number of different functions before they get they idea.

\begin{array}{l}\begin{matrix} y=f\left( x \right) & \quad \ y=4f\left( x \right) & \quad y=f\left( 4x \right) \\ \end{matrix}\\\begin{matrix} y={{x}^{3}} & y=4{{x}^{2}} & y={{\left( 4x \right)}^{2}} \\ y=\sqrt{x} & y=4\sqrt{x} & y=\sqrt{4x} \\ y=3x+2 & y=4(3x+2) & y=3\left( 4x \right)+2 \\ \end{matrix}\end{array}

Those functions whose reflections in the line x is also a function have an inverse and the inverse works the same for all these functions: linear, square root, cubic, exponential, etc. When you reflect in this line, the y‘s become the x‘s and the x‘s become the y‘s. As a result:

  • We have an algorithm for finding the equation of an inverse that calls for swapping the x and the y and solving for the new y.
  • The dimensions of the domain of the parent function become the dimensions of the range of the inverse function. The dimensions of the range of the parent function become the dimensions of the domain of the inverse function.

The y‘s become the x‘s and the x‘s become the y‘s. The range becomes the domain and the domain becomes the range. The second bullet is a direct consequence of the first. Inverse functions are so useful that we truncate the trigonometric functions in order to create them, and we actually have a totally artificial, man-made inverse of the exponential function, namely the logarithmic function.

Some functions have a y-intercept, an x-intercept, vertical or horizontal asymptotes, or periods and amplitudes. Some are algebraic and some are transcendental. Each is needed to model some relationship in the real world: periodic functions for sound waves or business cycles, exponential (or logistic) functions for population growth, etc.

In A Tour of the Calculus, David Berlinski presents and discusses the Mandela of the Functions. I share the image from this book with my students. Here is a link to a web site where you can see how the functions come full circle. Page down once to see the diagram.

The Mandala of the Functions

All of these are based on our Cartesian coordinate system, but there are functions whose equation are much simpler if expressed in terms of distance from the origin (pole) and direction from the positive x-axis (initial ray). Polar coordinates and the trigonometric form of a complex number have a lot in common, namely the Pythagorean Theorem. The work we have done on building the unit circle from the 45-45-90 and the 30-60-90 triangles sets the stage. This year, for the first time, my students knew the special right triangles by heart and so were easily to solve both of these types of problems.

As I wrote this I wondered why it was that I finally made these connections myself. Was it “just” another year of experience and reading books & blogs? I think I owe much credit to a summer course I took at a local college. For two weeks, a different very experienced teacher came in each day and shared their best lessons and ideas. It was an amazing experience and I still refer back to those notes frequently. Thank you, Art!

Making Sense of Quadratics

As a student, I never understood why we set quadratics equations equal to zero and factored to solve them. This really irked me. An article a few years ago in Mathematics Teacher inspired this set of Math Labs. I have different groups of students work on each of these and present the results, then provide each student with a photocopy of the completed set. Each group also puts their graph on a poster-sized sheet of graph paper, and these stay up on my side wall throughout the year. We frequently come back to them, to remind them of the concepts such as double roots that these illustrate.

Labs 1 through 4 are parabolas, Lab 5 extends the lesson to show that the product of three linear functions is a cubic.