Guide to Graph Plotting Programme
written by J Calder
_____________________________________________________________________________

Purpose       to help you learn about graphs of equations
	      by giving you a computerised way of trying out
	      lots of equations in a graph space
	      It is especially useful for looking at families
	      of equations like "straight lines" and "parabolas"
	      and seeing what happens as we change the numbers.

Access        from "Active Maths" startup screen
	      choose "Graph Plotting"
	      While in the graph space, you can call this guide on
	      to the screen anytime by pressing F4


Save typing   When you are studying a family of equations which are
	      all similar, you don't need to type equations again and again.

F9 key        the F9 key will COPY your previous equation and you only
	      need to change the detail you are studying.

or open edit  You can even arrow into a previous equation and change it,
	       


Equations     the programme types  y =  and you finish it off.

	      FOR EXAMPLE - STRAIGHT LINES

	      start with    y = x

	      then try equations where you add or subtract numbers
	      to the  x
			    y = x + 1
			    y = x + 4
			    y = x - 3
	      and so on




multiply      you can then clear your graph space with the F5 key
	      and try out   x  with multiplications and divisions

note    /     NOTE use of  /  for "divide" with this system


	      start with    y = x
	      then          y = 2x
			    y = 4x     look for what is happening
			    y = 6x     as the number gets greater...

			    y = 0.5x   ...and smaller

			    y = x/3    is  "y = x divided by 3"
			    y = x/5

	      "y = a half times x" looks like  y = 1/2 * x
	       which has the same meaning as   y = x/2
	       Try them both out.




2 ideas       Try 2 of these ideas together.  You may learn better if
together      you keep one number constant (means "the same") for a
	      while, like 3 or 4 equations, and change the other.

			     y = 2x + 1    keeping the  "2"  constant
			     y = 2x + 4
			     y = 2x - 3
			     y = 2x - 1

	      Clear the graph space with  F5  if it's getting too
	      full to see clearly what's happening, and make up your
	      own families of equations.
	      Look for ways of making sense of the patterns.

		   e.g       y = x + 4
			     y = 2x + 4
			     y = 3x + 4
			     y = x/2 + 4


	      Be sure to try these and your own equations like them:

			     y = -x
			     y = -2x
			     y = -5x
			     y = -1/2 * x



Parabolas     Begin your look at parabolas with the simplest equation:

			     y = x       "y = x to the power of 2"

	      You can see what multiplication by different numbers
	      does to the shape:
			     y = 2x 
			     y = 3x 
			     y = 4x     and so on

	      also           y = x  / 2
			     y = x  / 3   and so on


	      Going back to your basic parabola, use these equations
	      to work out how you can control the shifting of the
	      vertex in both  x  and  y  directions:
	      Remember to clear the graph space first
	      with the F5 key..

			     y = x 
			     y = x   + 3
			     y = x   + 5
			     y = x   - 2   and so on.  Get the idea?



			     y = (x+2)
			     y = (x+3)
			     y = (x+5) 

	      These last 3 equations should be shifting the vertex
	      to the left.  From them, work out the equations needed
	      to get a shift the other way, to the right.  Try them 
	      out.


2 ideas       Now try these equations which put the 2 shift ideas 
together      together:  You will probably need another Clear first!

			     y = (x+3)  - 4
			     y = (x-3)  - 4
			     y = (x-7)  + 3

	      Challenge :  make up and enter the equations which
	      shift the vertex to:
				 ( 8,-4)
				 (-5, 2)
				 (-1,-2)

	      And another new idea for you..

	      try this one!  y = -x  

	      What will happen when you enter the above equations
	      again but with  -  signs on them?  Try it out!


	      Other equations in the parabola family look like this:

			     y = (x + 4)(x - 2)
			     y = (x + 3)(x - 5)

	      Check them out and see how the numbers show up on the
	      graph.



Other curves  You can easily look at the "hyperbolas" which
	      start with the equation

			     y = 1/x

	      And the ideas you have been using here to move the
	      centre work on this one too.
	      Here is going up and down:

			     y = 1/x + 2
			     y = 1/x + 5
			     y = 1/x - 1
			     y = 1/x - 3


	      And of course, sideways shifting works by putting
	      x in a bracket with a number:

			     y = 1/(x + 4)
			     y = 1/(x + 7)
			     y = 1/(x - 3)
			     y = 1/(x - 5)


Higher powers  I have been able to make the computer read
	      "power of 2" for you from the F2 key as "  "
	      but you'll need the " ^ " mark to do other powers.
	      I've programmed this mark on to the F6 key.
	      It is normally [Shift] + [6]      

	      So to type "x to the power of 3" on this system you
	      need to go with   x^3

	      Some interesting higher power examples:  note that
	      these lines are usually very steep and it is easier to
	      get a good look at their shapes by mutiplying them by
	      small numbers like  0.2              

			     y = x^3         power of 3 curve
			     y = 0.2 x^3
			     y = x^3 + 4
			     y = x^3 - 2
			     y = (x + 4)^3   that sideways shift 
			     y = (x - 4)^3   again!

	      then clear with F5 and try these 2

			     y = 0.2x^3 - x + 3
			     y =-0.2x^3 + x - 3

	      and looking at the powers as a family:

			     y = 0.1 x
			     y = 0.1 x^3
			     y = 0.1 x^4
			     y = 0.1 x^5
			     y = 0.1 x^6
			     y = 0.1 x^7

	      notice how the odd and even powers compare.



Note for CGA  The older CGA graphics system does not have a    for
	      "power of 2" but the F2 key will still put
	      the  ^2  in for you.






Advanced      Ideas for teachers and advanced students

	      There are some built-in functions in our computer system
	      These include: (note the use of brackets)

			     y = sqr(x)     square root
			     y = sin(x)
			     y = cos(x)
			     y = tan(x)

	      also:   exp, log, atn (inverse tangent)                      

	      abs   - "absolute value" can be useful

	      cint  - "round off to nearest integer" does some
		       interesting things when thrown into your
		       equations


Circles       The standard equation  x + y = 1  for radius = 1
	      needs re-arranging to  y = sqr(1 - x)

	      Even then, you will only get the top half of a circle
	      because this system can only handle "functions" as in
	      only one y value for each x value.  The trick is to
	      enter the equation again with a  -ve sign on it.

	      Shifting follows the same pattern as for anything else:

	      eg with r = 3  y = sqr(9 - x)

	      and shift it   y = sqr(9 - (x-5) ) + 4   along 5 up 4

	      That is: to go along 5, replace 'x' with a bracket '(x-5)'
		       to go up 4 , add '+4' to the equation.



Trig          Works in radians in its present raw form.
	      I am working on a higher level programme that will offer
	      an alternative graph space for trig equations.
	      In the meantime, here is a re-scaling which makes good
	      use of the space for demo purposes:

			     y = 4 sin(0.79x)
			     y = 4 cos(0.79x)
			     y = 4 tan(0.79x)

	      The  x-axis becomes:  1 -> 45  ,  2 -> 90  ...


Simultaneous  A honey with this graph space; I've set it up for
Equations     multiple function display partly with these in mind.
	      Non-linear simultaneous equations are no problem!
	      You need to rearrange your equations into the form

			     y = x expression

	      then enter the 2 rearranged equations, and the
	      solution set is the  x  and  y  co-ordinates
	      of where the lines cross.


HELP          Some of the advice in this worksheet pops up on screen
	      as a HELP window when you press the F1 key
	      The F4 key brings this whole document on to the screen.


Error with    If the input equation does not follow the system rules
input         the present form of the programme will usually do nothing
	      and ask for the next equation.

	      You may also see a line plotted along the x-axis.

	      A common error is to use  \  instead of  /  for division.
	      The effect can be quite interesting and a good learning
	      opportunity if you want to experiment with it.
	      \  is "integer division" with rounding off of all values
	      involved.


Improvements  are possible!   What I'm now working on are:

		 *   alternative set of axes for TRIG with a
		     choice of working in degrees or radians

		 *   (further ahead) workspace for exploring
		     transformation of shapes, including use
		     of transformation matrices

		 *   re-scaling of axes, set your own range and domain

	      All suggestions are welcome, and I hope it's useful! 


Contact me    John Calder, Box 41-076, Auckland 3, ph 8282612
