4. A tour of functions
Section author: Leina Gries <leinagries@gmail.com>
[status: content-mostly-written]
4.1. Motivation, Prerequisites, Plan
A function is a relationship between two (or more) variables. For now we will call one of them \(x\) and the other \(y = f(x)\) (the value of the function at x). For this relationship to be a true function, each value of x must give a unique value of y.
We use other terminology for functions. We can say:
y is a function of x
f is a mapping from x to y
There are other terms people use as well. As usual, don’t worry about differing terminology: they mean more of less the same thing: you take a value x, put it into a machine, crank the machine, and out pops the value y.
The goal of this chapter is to gain a deeper understanding of mathematical functions and how to go about plotting them. Initally, we will experiment with basic linear functions, using them as a tool to understand plotting software and seeing how visualization can be useful. We will experiment with plotting multiple lines at a time, and look at how the geometry matches up with the algebra.
Then we will look at polynomials and explore how they solve some physical problems and how the solutions to these problems can be visualized with plotting software.
Afterwards, we will move into the realm of more advanced functions- touring exponential growth and decay, the curves made by sine and cosine waves, bell curves (Gaussian Distribution), factorials and the fibonacci sequence. Then, we will cover some functions that create interesting and visually pleasing graphs.
Finally, we will apply our findings to the real world and plot functions that describe real life events, such as climate change and nuclear fission.
4.2. Linear Functions
To start, let’s stay linear. Graph the equation \(y=5x-8\) by running gnuplot and typing the following expressions:
gnuplot> set grid
gnuplot> plot 5*x - 8
This should bring up a graph displaying the equation in a seperate window. Try plotting this with and without grid lines, using the command (code). Explore the window. What can you do to make the line displayed change?
Can you make the slope negative?
Can you graph several lines at once? This brings us to plotting two linear functions. Try plotting the original line with the addition of \(y=4x-6\).
gnuplot> set grid
gnuplot> plot 5*x - 8
gnuplot> replot 4*x - 6
Figure 4.2.1 Two lines, one with slope 5 and the other with slope 4. Lines with different slopes will always intersect.
We see visually in Figure 4.2.1 that the lines intersect. How do we figure out at which point they touch? Visually we can see that x is near 2 and y is near 2, but can we get the exact values?
This is where the beauty of analytic geometry comes in: there is a relationship between the gometry (which we see visually) and the analysis that you can do using algebra. The key here is:
The intersection is the point at which the x and y values of both
lines are the same.
How do we express that algebraically? We can start by saying that if \(y = 5 x - 8\) and \(y = 4 x - 6\), then:
There are many ways of solving this, and depending on where you are in your math studies you might have seen one or more. I will do one of the simplest: if \(5 x - 8\) and \(4 x - 6\) are equal, then I can add and subtract the same quantities from both, and the result will still be equal. That gives
Once we have x we can plug that value into either of the equations for y and we get \(y = 2\), so the intersection point is \((2, 2)\)
This technique can be used in many other circumstances to find where two curves intersect. The algebra will be more complicated than finding the intersection of lines, but the idea will be the same.
Can you think of a way to find out multiple points of intersection between three or more lines?
Another bit of terminology: these functions in which x appears only in the first power (there is no \(x^2\) or \(x^3\)) always give straight lines when you plot them. Because of this we call them linear functions.
FIXME: discuss Direct proportionality.
4.3. Polynomials
Polynomials are the next step in complexity. Let us start with a binomial (second degree polynomial). Plot the function \(y=3x^2+6x-4\)
Figure 4.3.1 A simple second degree polynomial \(y=3x^2+6x-4\)
From the graph, what can you observe about this function? Where are the x intercepts? The y intercept? While these are clear in some cases, they are difficult to find in others.
The figure that comes from plotting a second degree polynomial is called a parabola. This one has two arms that grow up from its minimum.
We will spend a bit of time discussing this plot because it teaches us some interesting things to look out for.
This polynomial has three terms: \(3 x^2\), \(6 x\) and \(-4\). Each of these three terms has more or less importance according to the value of \(x\). When x is 0, then the constant term 4 dominates. When x is between 0 and 1 or 0 and -1 then the term \(6 x\) grows to be more important. When x gets much bigger than 1 (or much smaller than -1) then the \(3 x^2\) term (second order term) dominates.
Terminology: the number that multiplies \(x^2\) (in our case 3) is called the coefficient of the second order term (or the quadratic term). The number that multiplies \(x\) (in our case 6) is called the coefficient of the first order term (or linear term). The term that doesn’t have x is called the constant term or the offset.
The arms of the parabola eventually point up. This is because the term with \(x^2\) gets much bigger than the other ones, and when x is negative, \(x^2\) is still positive.
Zero crossings: it is interesting to note where the parabola crosses the x axis and the y axis. In class we discuss how to find this by zooming in on the picture (both with the mouse and with the range parameters in gnuplot). Terminology: the values of x at which a function crosses the x axis (i.e. where \(y = 0\)) are called the roots of the function.
It is also interesting to note the value of x for which the parabola has its minimum or maximum values. Sometimes there is no global minimum or maximum, so in class I describe what a local minimum or maximum is. There is a technique for finding maximum and minimum values for a function. This involves calculating the derivative of the function. We will say a bit more about derivatives below.
4.3.1. Derivatives
Without introducing the entire subject of differential calculus (for that see Section 15), let’s look at the parabola we have been considering: \(y=3x^2+6x-4\). Looking at the plot, study what the slope of that plot is for big negative x, for x between -3 and 0, and for big positive x.
We expect the slope to be big and negative for very big negative x, close to zero for that value of x between -3 and 0, and then to get positive and big.
When we learn calculus we will learn that you can figure out exactly what this slope is at every point in the curve. Without showing how it’s done, I show the result here. We call the function that is the slope of \(f(x)\) the derivative of f, and we write \(\frac{df(x)}{dx}\). The derivative of this second degree polynomial is:
The properties of derivatives that we used here were: (a) the derivative of a sum of terms is the sum of the derivatives of each term, and (b) the derivative of \(x^n\) is \(dx^n/dx = n \times x^{n-1}\).
The place at which the derivative is zero is \(x = -1\), so we can say that the our parabola has its minimum at \(x = -1\). If you zoom in to the plot enough you will see that this is true geometrically.
You should explore this polynomial further. You should change the sign or some of the values and see how different the plot looks.
4.4. Higher order polynomials
Having successfully plotted a binomial, let’s move onto higher order polynomials. Plot \(x^4 + x^3 - 7 x^2 - x + 6\). The result will initially look somewhat boring.
Figure 4.4.1 A fourth degree (quartic) polynomial \(y = x^4 + x^3 - 7 x^2 - x + 6\). This does not look too interesting at this time.
Figure 4.4.2 The same fourth degree (quartic) polynomial \(y = x^4 + x^3 - 7 x^2 - x + 6\). By zooming in to the x range [-3.2:3.2] we see interesting features.
Now, finding the root (or solutions, found at the x intercepts of the equation) are much more dificult to find manually.
By zooming in further, find the roots of this 4th order polynomial. How many are there? Does this relate to the degree of the polynomial?
And now let me tell you how I picked this polynomial: I wanted to find a polynomial that crosses the x axis at -3, -1, 1 and 2. How do you do that? First notice that the first order polynomial \((x-1)\) is zero when \(x = 1\). So we can craft a set of first 1storder polynomials (linear functions) which cross the x axis at -3, \((x + 3)\), -1 \((x + 1)\), 1 \((x -1)\) and 2 \((x - 2)\). Since zero multiplied by any number gives zero, we can try to multiply these together. We get:
If you multiply out all the expressions in parentheses you will find that:
Would you have been able to go the other way: go from \(x^4 + x^3 - 7 x^2 - x + 6\) to \((x + 3) \times (x + 1) \times (x - 1) \times (x - 2)\)? It’s much harder to do, and it’s called factoring a polynomial.
The factored expression is useful because it tells you clearly where all the roots of the polynomial are. The normal expression is useful because you can see clearly the coefficients of the terms of each order. In particular, the fact that the \(x^4\) has a positive coefficient (in this case 1) and an even power allows you to say that the curve will have arms that point up on both sides.
Another bit of terminology: the zoomed plot in Figure 4.4.2 shows a lot of structure near \(x = 0\), but the unzoomed plot in Figure 4.4.1 shows that for large negative and positive values of x you just go up. The behavior of a function when you go to large values of x (or large negative values) is called asymptotic behavior, and understanding the asymptotic behavior of a function is a useful and deep area of mathematics.
Note that with simple polynomials, it is relativly easy to graph manually and to find the y for a given x. However with higher order polynomials a plotting program helps us gain insight.
4.5. Inverse functions
FIXME: must write; discuss 1/x and x^-n and x^-(a/b)
Mention inverse proportionality.
When one
4.6. Elementary transcendental functions
A function which cannot be written as a polynomial is called a transcendental function.
4.6.1. Exponentials
Exponential growth and decay are commonly seen phenomena in daily life. The exponential function is be used to model many real life situations with growth and decay.
4.6.1.1. Growth
Let us start with a simple problem and a small program to simulated it. Let’s say we have a pair of rabbits, and every month they produce two baby rabbits. (NOTE: rabbits do have a one month gestation time, but their litters are actually much bigger – 4 to 12 kits, but we’re keeping it simple).
After one generation you have 2 new rabbits (a total of 4). Then those pair up and reproduce and you have 4 new rabbits (a total of 8).
A simple program to demonstrate this is:
#! /usr/bin/env python3
import math
def main():
n_generations = 10
n0 = 2 # initial population
print('## generation n_rabbits')
# print('## generation n_rabbits exponential')
n = n0
for generation in range(1, 11): # let's do 10 months
n_new = (n / 2.0) * 2
n = n + n_new
print(generation, n)
# print(generation, n, math.pow(2.0, generation+1))
main()
Run this program and notice that the
The general formula for exponential growth or decay is written \(A(t) = P_0e^{r*t}\), where A is the amount at any given t, time, r is the rate of growth or decay, and \(P_0\) is the initial amount of whatever is being changed. It is important to remember here that e is simply a number – it is not a variable, and should remain constant in each situation. In this way, it should be treated the same as pi.
Let us explore this function:
Wait, wait wait: what does this even mean? We have seen what it means to take a number, like 4, and raise it to a power, like 3: \(4^3 = 4 \times 4 \times 4 = 64\). But what does it mean to take numbers that are not integers and raise them to powers? That’s straightforward if the power is an integer. For example: \(1.7^3 = 1.7 \times 1.7 \times 1.7 = 4.913\). That’s all fine.
But what does it mean to take a number to a power that is not an integer? The topic is too long to go in to in detail, but remember this property you probably learned a long time ago:
This allows us to define fractional exponents:
which means that \(x^{1/2}\) is that number which, when squared, gives x, which is the very definition of square root! Thus:
Using the properties of exponents that we know, this allows us to get pretty close to almost all expressions \(a^b\) where a and b are real numbers, not just integers or fractions.
The number \(e\) is a special number, called Euler’s number. It’s numeric value is about 2.71828, and it comes up in so many places in math that it is as important as :math:pi`.
Let us look at our rabbit reproduction problem and write out an extra number. Change the line that prints the current population to say:
print(generation, n, math.pow(2.0, generation+1))
Note that you will also need to import math at the top of your
program.
If you run the program again you will see that the exponential function \(2^(generation+1)\) matches what our simulation of rabbit growth gives.
At some point you will learn that you can convert an exponential in base e to an exponential in base 2, or 10, or any other number. The three important ones are e (occurs naturally in math), 2 (comes up in computer science) and 10 (we have 10 fingers, and our digits are 0 to 9). This allows us to write:
FIXME
Introduce rabbit death. Then think about introducing foxes. Refer to the proposed chapter on predator-prey problems.
4.6.1.2. Decay
Uranium 235 has a half-life of 703 million years. Let’s write a program to simulate how some \(5 \times 10^22\) atoms of U235 would decay.
#! /usr/bin/env python3
def main():
n_steps = 10
## U235 density is 19.1 g/cm^3
n0 = 5e22 # initial population
print('## time_yr N_U235')
n = n0
time = 0
## U235 half life 703.8 million years
while time < 7.03e10: # 100 half-lifes
n = (n / 2.0)
print('%10.4g %10.4g' % (time, n))
time += 7.03e8
main()
Run the program and discuss what the meaning of half-life is and how it relates to exponential decay.
4.6.2. Trigonometric function
4.7. Gaussian distribution
Gaussian distribution, more simply known as the bell curve or normal distribution, is found in many places in the world around us. When the heights of children of a similar ages are graphed in relation to one another, the result is a bell curve. Named for its distinctive, bell like shape, the bell curve can be easily graphed with knowledge of a relatively simple equation. It is :(equation-formatting?) where () is the mean, and () is the standard deviation.
Now, of course we could calculate these values manaually, but why would we?Use (code) to generate a random set of numbers. Start out with about 20.
Now, use () to find the sum of these numbers. Check your answer logically- does this seem accurate? Now, divide this by the number of values you used.
This works, but it would be easy to cut out the middle step- use (code) to determine the mean with no possibility for user error.
Next, you can find the standard deviation- use (code).
Now, you’re all set up to plug these numbers into the equation listed above. But this seems time consuming, and programming is all about finding ways to minimize repetitive tasks. Use (code) to find the standard deviation of your data set.
Try it with data for the heights of your school, or for some other large data set that you find interesting.
Next, plot this data using (code).
Try manipulating the graph using the above techniques.