Relationships are a huge part of the secondary school experience. You form all sorts of relationships with people in your school. In English class you study Romeo and Juliet, in social science classes you often learn about Canada’s relationship with other countries, in health class you learn about the human body’s relationship to diet and nutrition. Math class is no different, we learn about relationships between variables. Steven Strogatz mathematics professor at Cornell University and author of “The Joy of x” says that high school level math essentially involves one of two activities, solving equations and working with formulas. The arithmetic you learned in elementary school becomes algebra when you start working with unknowns and formulas.
A “function” is essentially a way to describe the relationship between two variables x, and f(x). The value of f(x) “depends” on the value of “x”. You started working with functions in grade 9, when you looked at the equation of a line … y = mx + b or f(x) = mx + b. For these functions x and f (x) have very straight forward relationships. As x increases in value f(x) either increased or decreased (has a positive slope or a negative slope). To make matters even more simplistic f(x) increased or decreased at a constant rate. Here is an example of a typical linear function, f(x) = 0.5x + 2 (displayed as a graph and table of values) that has a positive slope.
In grade 10 things got slightly trickier as you tackled “quadratic functions”. Quadratic functions are degree 2 functions such as f(x) = or f(x) = ( x + 1) (x + 2) = . In these functions x and f(x) have a more complicated relationship with each other! Take for example the function f(x) = , when x is less than zero … as x increases the values of f(x) decrease, at x = 0, F(0) = 0 then things start to change when x is greater than zero … as x increases the values of f(x) increase. To make things a bit more complicated the rate at which f(x) decreases or increases is never constant. When you graph this relationship you get a curved shape known as the “parabola”.
In grade 11 things became quite mind boggling as you explored the properties of exponential functions which grow or decay at unimaginably fast rates, and trigonometric functions which behave like waves that repeat themselves over and over again.
In Grade 12 you we will be looking at Power (or Polynomial) functions. Power functions are of the form f(x) = , where a variable x is raised to a fixed power n. Linear functions (when n = 1) and quadratic functions (when n = 2) are classified as Power functions. Notice that a quadratic function is the product of linear functions. For example F(x) = (x + 2) (x + 3) = or F(x) = (2x -1) (x + 4) = . What happens when we multiply 3 linear functions with each other? 4 linear functions? What do their graphs look like?
Now let us look at a Power function relationship that is relevant in the real life! You might have a parent or adult relative that invests in their money in the stock market. People invest in the stock market because of the potential return on their investment is significantly higher than simply leaving your money in a savings account or GIC. However there is risk involved as well, there is no guarantee that your investment will yield a return … sometimes you incur a loss. Suppose that your investments drops by 50% (which means the value of your investment is half of what it was originally) and then gains 50% the next. One might think that you are back where you started … after all you just got back the 50% you lost. Unfortunately this is not true. A 50% gain multiplies your money by (1 -0.5) = 0.5, and a 50% gain multiplies it by (1 + 0.5) = 1.5. When those happen back to back, your money multiplies by 0.5 x 1.5 = 0.75 or in other words you are still down 25%!
With the analysis of Power functions we can understand why, you will never get back to even when you lose and gain the same percentage in consecutive years. The Power function that models our situation is …
F(x) = (1 – x) (1 + x) = 1 – x2,
where x is the percentage lossed/gained. In the down year your investment shrinks by a factor 1 – x (where x = 0.50 in the example above), and then grows by a factor 1 + x the following year. So the net change is a factor of (1 –x )(1 + x) and if you expand equals 1 – x2. The point is that this expression is always less than 1 for any x value other than 0. So you never really recoup your losses.