Naming and Graphing Functions

Date: 06/16/97 at 09:50:30
From: Perfect Distributors
Subject: Functions

Dr. Math,

I have been given an assignment on functions and I have no idea where 
to start. In the first part of the assignment, we have to name seven 
functions:

a) y = a(x-b)^2+c
b) y = ax^2+bx+c
c) y = ab^cx+d
d) y = (a/cx)+b
e) y = (a)(sin)(bx+c)+d
f) y = (a)(cos)(bx+c)+d
g) y = (a)(tan)(bx+c)+d

Then we have to describe the effect on the graphs of these functions
when selected constants a, b, c or d are altered.

Can you help point me in the right direction?

- Ben

Date: 06/16/97 at 12:52:00
From: Doctor Mike
Subject: Re: Functions

G'Day Ben,
  
In this country we have an expression to "separate the wheat from the 
chaff", which means to ignore the distractions and get down to the 
important stuff.  We all speak English, but I never know for sure 
about a particular word or phrase.
  
In each case you have y as a function of x, and the constants give 
many many different variations on the basic theme. To see the main 
idea, look what happens when the constant multipliers are equal to one 
and the constants that are added or subtracted are equal to zero.  

In the case of the function in (a), you get y = 1(x-0)^2+0 = x^2, or 
y equals x squared. So, the name of the basic family of functions in 
(a) is the "squaring" function. The one in (b) is related to it, but 
is a bit more general, namely the "quadratic" functions.  
  
The function in (c) is a bit puzzling. The normal convention is to
give exponentials a higher priority than multiplication, so this would 
mean "y = a(b^c)x+d" which would be variations on "y = x". However,
I suspect your source meant "y = ab^(cx)+d" which would be variations
on "y = b^x". So, the name for this family could be "identity" or
maybe "exponential".  
   
For (d) the basic version is "y = 1/x" which is the "reciprocal" 
function.

The last three are the trigonometric "sine" "cosine" and "tangent" 
functions.
  
Okay, now that we have names for the function families we are dealing
with, let's see what kind of variety we can find in these families.
I'll give just a few details on one of them to start you off. You 
will need to spend a lot of time with this to really get a feel for 
what is happening (and go through most of a pad of graph paper to 
boot!)
    
   Theme      y = sin(x)            for x = 0 to x = 360 degrees 
   Var. 1     y = 2*sin(x)          for x = 0 to x = 360 degrees
   Var. 2     y = sin(2*x)          for x = 0 to x = 360 degrees
   Var. 3     y = sin(x)+1          for x = 0 to x = 360 degrees
   Var. 4     y = sin(x-90)         for x = 0 to x = 360 degrees
   Var. 5     y = 2*sin(2*x-90)+1   for x = 0 to x = 360 degrees
    
I assume you are familiar with the basic shape of the sine function.
It starts with sin(0) = 0 and the graph makes a "hill" and then a
"valley" and winds up again with sin(360) = 0. The high point is at
sin(90) = 1 and the low point is sin(270) = -1. With me so far? 
   
Variation 1 changes the height of the hill (twice as high) and the
depth of the valley at 270 degrees (twice as deep) by using the 
multiplicative constant. Graph it. This is a "hands on" math lab.
This is sometimes called an "amplitude" variation.
   
Variation 2 is a little tricky. As x goes from 0 to 360, 2*x goes
from 0 to 720, or twice around the circle. The graph, then, has a
"hill", a "valley", "another hill" and "another valley" before x gets 
to 360. This is sometimes called a "frequency" variation. In this 
case, the hills and valleys are "more frequent" for your trip. 
Definitely, you want to carefully sketch the graph of this one.
  
Variation 3 just adds one (1) to each and every y-value. This raises
the graph vertically by 1 unit. The result here is that it now looks 
like the bottom of the "valley" rests on the horizontal axis.
   
Variation 4 changes the angle you take the sine of, e.g. if x = 0 you
take the sine of -90 degrees. If x = 100 you take the sine of 10
degrees. If you go out to the maximum value of x, namely x = 360, for
particular function, then you are to evaluate y = sin(270). You are 
using the same basic "sine" function, but the "-90" part shifts the 
range of angles you are using. Sometimes this kind of thing for a trig 
function is called a "phase shift" variation.
   
Variation 5 is a combination of all the others. It has amplitude,
frequency, and phase shift variations, and an increase in value. Thus 
it has all kinds of variation applied at the same time.  
  
I hopes this helps, but you will have to try a lot of variations on
your own before it really sinks in.  

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 06/18/97 at 02:22:53
From: Perfect Distributors
Subject: Functions (again)

Dr. Math,

Thanks for your help before. I have attempted to draw graphs of the 
first two functions to find what each variable does. I have discovered 
what most variables do, but was wondering if you could help me name 
what each variable does (such as phase shift in the case of trig 
functions). I am still unclear on two: y = ab^(cx)+d and y = (a/cx)+b. 
I couldn't isolate the indervidual variables to see their results on 
the graphs of these two functions.

Once again, 
Thanks from Ben

Date: 06/20/97 at 20:31:34
From: Doctor Mike
Subject: Re: Functions (again)

Hi again,   

Let's handle the easy part first. The "+d" or "+b" at the end just 
moves the graph up or down. Think of putting a piece of graph paper
on the table and a transparent sheet on top of it. If you then draw 
the graph, it will appear that you are drawing on the graph paper, but 
what you drew is really on the transparent sheet. You can then see 
the effect of adding/subtracting a constant by sliding the transparent 
sheet up/down. (Replacing "x" by "x+k" or "x-k" is like sliding the 
sheet left/right.)
   
Next, the "a" multipliers are like the amplitude effects I mentioned 
last time.  Larger multipliers make any hills higher and the valleys 
deeper, and smaller multipliers make them not so high and not so deep.  
However, the word "amplitude" is not commonly used except for the 
"wavy" functions like sine. The two types of functions do not have 
hills or valleys, but the multipliers do tend to stretch or collapse 
their shapes up and down. Note that in "y = (a/cx)+b" it could also be 
written "y =(a/c)*(1/x)+b" so the parameter c is really part of the 
multiplier. The larger c gets, the smaller the (a/c) multiplier gets.
   
If we eliminate the things mentioned above from the exponential 
function, we get y = b^(cx). These two constants, the base "b" and 
the coefficient "c" of x, affect the size of the graph, but the result 
will always be sort of a "swoosh" in one direction on another. You 
can see the two basic shapes by doing the graphs for "y = 2^x" and 
"y = (1/2)^x. As you compute larger and larger "x" exponents of 2, the 
results get larger and larger (2, 4, 8, 16, 32, 64, 128, .....). As 
you compute larger and larger "x" exponents of 1/2, the results get 
smaller and smaller (1/2, 1/4, 1/8, 1/16, .....). What happens on the 
left end (negative x) of these two graphs?  You can see where the 
graphs cross the vertical y-axis by setting x = 0 and solving for y.  
What do you get?  
  
It's difficult to talk about these things without drawing a few, so 
you will have to do that. Pick out some particular members of these 
families of functions and graph them to see what they look like.  Good 
luck. I hope this helps some.   

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/