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Locus

Another exclusive Foundation RISCWorld application

Exercise 4. Constants and Families

Many graphs are related to each other and Locus lets you study the effect of changing one or more constants in your formula definitions and to plot several graphs at one time.

This Exercise will describe one example and then you can experiment for yourself.

If you are studying the shapes of simple quadratics you might be interested in the effect of changing the coefficient of x.

1. Turn off auto-scaling
2. Set the axes as follows:

1. xmin: -7

1. xmax: 5
   
      ymin: -10
   
      ymax: 30
2. Click on <Plot> to set the axes
3. Type x²+bx into the formula input icon.
4. Click on the constants button. (This is the one with 'abc' written on it.)
5. Type 0 into the second field, marked 'b'
6. Click on <OK>
7. Click on the Family button which is to the right of the constants button.
8. Type 5 into the field at the top (This sets the number of graphs to be drawn.)
9. Type 1 into the 'b' field on the increment pane (This sets the amount of the increment.)
10. Click on <OK>
    
    

It is acceptable to set several constants and change them simultaneously if you wish, although this is normally regarded as bad practice.

As an example, let us suppose you are doing some work on projectiles. It is possible that you will want to set several constants at the start and then alter them one at a time.

In order to examine this we need to consider how Locus deals with parametric equations. If you already understand parametric equations you may prefer to skip the following paragraph.

A pair of parametric equations is a mathematical construct where, instead of the more familiar case where we define y in terms of x, we define both x and y in terms of a third variable (in this case 't' which represents time.) When considering the motion of a ball thrown into the air, the standard cartesian equation of the path is very awkward to use and does not contain all of the information that we need, for example, it makes no mention of time. By using parametric equations we are able to define the path and analyse the motion taking time into consideration.

As an example.

1. Click on the Clear Screen button
2. Turn off auto-scaling
3. Make sure you are in 'degrees' mode (the full circle is 'on')
4. Click on the button on the top row of the main window pane which contains the letters x,y,t
5. Set the axes as follows:
   
      tmin:0
   
      tmax:4
   
      xmin:-2
   
      xmax:50
   
      ymin:-5
   
      ymax:30
6. Click on <Plot> to set the axes
7. Select the equation set utcos(a),y+utsin(a)-9.8t² on the Formulae menu

There are two things you should take note of at this point. Firstly, both the x and y equations are written on the same line separated by a comma. Secondly, if you are finding trigonometric functions of constants Locus requires that you put parentheses around them.

1. Click SELECT on the constant ('abc') icon
2. Set a=5;u=25;y=0 and click on the <OK> button
3. Click on the family icon next to it
4. Type 9 into the 'Number of Graphs' icon.
5. Type 10 into the 'a' increment icon
6. Click on <OK>

You should see a series of negative quadratics which get progressively higher but d not travel as far horizontally. You should note which angle gives the maximum horizontal range.

As an exercise try using the same equation set, but this time set the value of a to 45 and only increment the value of y. Draw seven graphs incrementing y by 1 each time. Using different values of a and y is a good way of demonstrating why an Olympic shot putter does not release the shot at 45 degrees to the horizontal.

Foundation RISCWorld