Using AutoGraph
A Self Help Guide
Getting Started
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Entering Equations
Setting Up Axes
Basic Plotting Options
Using Constants
Drawing Gradient Graphs
Using The Objects Menu
3D Graphs
Statistical Graphs
Entering Equations 1
You can enter equations in a range of formats
•explicit (e.g. y = x2)
•implicit (e.g. x2 + y2 = 9)
•parametric: (e.g. x = cos t, y = sin t)
•differential: (e.g. d2y/dx2+3dy/dx = -3y)
•polar: (e.g. r = cos q)
To enter an equation click on the “Enter Equation” button
through the menu “Equation” : “Enter Equation…”
or go
You should end up with the dialogue box on the next page…
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Entering Equations 2
If you want, you
can give your
equation a
meaningful name
like “Quadratic 1”
Type your equation
or inequality in
here. You can use
constants in your
formulas to
investigate how
they affect the
shape of the graph.
Use these buttons to
quickly include
powers etc.
Note : alt 2 gets you 2
and so on.
If your equation contains constants they
are automatically set to 1, click here to
change their values now, or use
CONSTANT CONTROLLER after the
graph is drawn.
Normally the graph is drawn
over the full range of x values
shown on the axis. Click here if
you want to restrict the x values
that the graph is drawn over.
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Change the colour and
thickness of the drawn graph by
clicking here.
Note : AutoGraph automatically
draws each new graph in a
different colour.
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Entering Equations 3
It is also possible to define two functions f(x) and g(x) which can then be used to
create graphs such as y=f(x), y=f(x+2) and so on.
You can enter functions by clicking on the “Function Definition…” button
gets you a simple dialogue box.
which
Simply enter the function definitions in the relevant boxes using alt 2 for squaring and
so on. Alternatively, copy the body of the function from the “Enter Function” or “Edit
Function” dialogue boxes and paste into this dialogue box (this allows you to enter
more complicated functions quickly.)
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Setting Up Axes 1
To set up axes to suit the graph you want to draw you need to click on the “Edit Axes…” button
or go through the menu “Axes” : “Edit Axes…”
There are also other buttons that can be used to change the axes in specific ways.
The buttons that are of most use are.
Default Scales : This sets the axes to what AutoGraph defines as a basic set, which is x
from –6 to 6 and y from-4 to 4 for most functions, and x from p/2 to 2p
and y from –2 to 2 for trig functions.
Equal Aspect Mode : This changes the x axes so that the spacing is equal to that of the
y axis. This ensures that circles appear as circles (not as ellipses) and
the line y = x is at 45o to the axes.
If you click on the
button you get the following dialogue box…
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For other tabs, see
later pages.
Setting Up Axes 2
Use these to change
how often numbers
are put on the axes
and how often
additional ‘tick’ marks
(called pips) are
drawn. You need to
deselect Auto before
you can change these
values.
Enter your choice of x
and y values here.
Sets the x and y axes
to the default settings.
Note: If you leave
Auto on, the values
are only updated
when you close this
dialogue box.
Changes the y axis to
suit the function you
are drawing and the
range of x values you
have chosen.
Changes the x axis spacing to
match the y axis spacing ensuring
circles appear as circles.
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Changes the y axis
minimum and maximum to
match the x axis.
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For other tabs, see
other pages.
Setting Up Axes 3
Use these to add a
fuller description of
the variable to the
axis.
You can change the
letters used on each
axis to match the
work you are doing.
For example, you
might want to use
(t,s) or (t,v) instead of
(x,y) if you are doing
a Mechanics question
For example, if you
have chosen to use
(t,s) as variables you
could then have
“Time” and
“Displacement” as
labels.
You would use s and t
in the formulae you
type in, and the words
“Time” and
“Displacement” would
appear on the axes of
your graph.
The default labels are
x and y.
Don’t worry about this (I don’t know what it does, and
it isn’t mentioned in the HELP stuff!)
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For other tabs, see
other pages.
Setting Up Axes 4
You can have no
axes, both axes or
just the horizontal
axis.
This allows you to put
arrows on the ends of
your axes.
You can change the
appearance of the grid or
remove it altogether. This
does not affect the
graphs (you can draw a
polar graph on Cartesian
axes.)
You can place the axes
at the bottom and left
hand edge of your
graph instead of
having them cross at
(0,0)
This allows you to
specify how often grid
lines are draw
between the pip marks
(see Ranges)
This changes the shape
of the graph from a
rectangle to a square
rescaling the axes or
cropping them (if you are
in equal aspect mode)
and leaving a blank
section.
This allows you to change where the equations are displayed and how many are displayed, but not the size of
the font used. Use the “None” option if you want to prepare examples in advance where you want students to
work out what the equation is, so it is important that the students can’t read the equations
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For other tabs, see
other pages.
Setting Up Axes 5
Use these to change
the colour and
thickness of the axes
and grid lines of your
graph and the colour
of the graph paper.
Use these to change
the style and size of
the font used on the
axes labels (if you
have them) and
numbers.
There are some
useful (and eyeboggling) styles of
graph paper
already set up.
Have a look at
them!
Use this to change
the size of the font in
the key. You can also
change the colour of
the box the key
appears in!
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Basic Plotting Options 1
One useful feature of AutoGraph is the ability to plot graphs slowly, pausing the
plotting and restarting it whenever you wish.
Pausing the plotting allows you to get students to predict what will happen next and
then check their ideas.
The key buttons are…
Slow Plot : Click this before you enter the equation to see it drawn slowly.
Click after the equation has been entered and then Replot to get the same
effect.
Pause : Allows you to stop and restart the plotting. You can also do this by
pressing the SPACE bar.
Replot : Use this to redraw a graph that has already been completely or
partly drawn.
Fast Forward : Allows you to quickly finish off drawing a graph once you
have gone past the points you were particularly interested in.
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Basic Plotting Options 2
Having paused your graph you can then get students to predict what will happen next,
and get them to show this on the graph using the following tools.
Point Mode : This enabled you to plot points on the graph (unfortunately as large
square blobs not crosses!)
Enter Co-ordinates : Use this if you want to type in the co-ordinates of a point
rather than plot it manually.
Scribble : Use this to draw freehand lines and curves over your graph.
Erase : Use this to rub out the freehand lines that you’ve accidentally drawn in
the wrong place!. Just click anywhere on the drawn line. Be careful, it also rubs
out graphs if you click on them!
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Using Constants 1
You can include constants in the equations you enter and explore families of curves
by varying the value of the constant using the CONSTANT CONTROLLER.
To get the constant controller on screen you click on the
button.
You then get the following dialogue box.
Gives you access to
other options for
behaviour of constants,
including animation.
See next page.
If you have more than
one constant, click here
to get access to the one
you want to change.
Gives the current value
of the constant.
Tells you how much the
constant is going to be
changed by.
Starts/Pauses
animation of currently
selected constant.
Pauses animation of all
constants.
Use these to increase
or decrease the step
size by a factor of 10.
Use these to increase or decrease
the constant by the indicated step
value.
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Using Constants 2
For the selected constant only…
You can set the constant
values yourself
Plot several graphs with
different values of the
constant.
Show how changing the
constant affects the graph
in an animation.
If you select Family plot or
Animation you can control
the range of values the
constant takes by specifying
a start and end value and
either the step size or the
number of values you want
to show.
You can also enter a list if there are
particular values of the constant you wish
to consider.
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Drawing Gradient Graphs 1
Once you have drawn a graph you can then add a gradient graph by clicking the
“Gradient Function…” button
this gets you the following dialogue box.
You can change the name of the
gradient function if you wish.
This tells you
which function you
are finding the
gradient of.
Click this to get
access to change
how the gradient
graph is drawn.
Click this to
change the colour
and style of the
gradient plot.
More details on
the next page.
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Drawing Gradient Graphs 2
Probably best to leave this
set to Automatic. Manual
should give you the option to
plot only part of the gradient
graph.
Draws the tangent at each
point on the curve. Moves as
the x value changes.
Pauses the plotting at
maxima and minima, use the
space bar to restart.
These options only work if
you switch on SLOW
PLOTTING (See Basic
Plotting Options 1)
Pauses the plotting at points
where the gradient is at a
maximum or minimum, use
the space bar to restart.
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Drawing Gradient Graphs 3
As well as drawing tangents when you are plotting the gradient graph, you can draw
tangents and normals to a curve at any point on the curve.
To do this click on the “Point Mode” button
and then place a point on the curve (as
you move the mouse close to the curve, the cursor changes from a small x to a arrow
indicating that the point will be attached to the curve.
Once you have a point on the curve you can go back to “Select Mode”
sure that only the point is selected.
and make
Clicking on the object menu (or right clicking on the point) gives you a range of options
including drawing a tangent, a normal, a vertical line through the point and so on.
If you want to derive your own gradient graph you could calculate the gradient of the
tangent at various points on the curve, plot the set of points (x, gradient) and try to fit a
function to your points.
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Using the Object Menu 1
If you select one or more objects (graphs, points etc) and right click (or use the object
menu) you gain access to a range of useful features.
Selecting multiple objects can be done by clicking on the first, then holding CTRL and
clicking on any others, or by switching on “Whiteboard Mode”
which allows you to
select multiple objects by clicking on them one after another (be careful though, you
often end up with multiple items when you only want one – to cancel your whole
selection simply click in an empty part of the workspace and to remove a single
selected object from a set of selected click on it.)
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Using the Object Menu 2
With a graph only selected you can…
Approximate the area under the graph using rectangles (under and over the curve, but
not mid-ordinate rule), trapezium rule and Simpson’s rule.
Find a root using interval bisection or Newton-Raphson.
Show the roots.
Show a table of x, f(x) values.
Turn the curve into a series of points (Use Create Dataset…)
Show how successive Maclaurin approximations fit the curve.
Draw evolutes.
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Using the Object Menu 3
With a point attached to a graph selected you can…
Draw a line through the point (i.e. a tangent, normal, horizontal, vertical or other line
with given gradient)
Draw a vector based at the point, or circle centred at the point.
Move the point to the next root or turning point.
Perform a Newton-Raphson iteration using the selected point as the start point.
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Using the Object Menu 4
With two points selected you can…
Draw a straight line through the points, or a segment between them.
Draw a vector from the first selection to the second.
Find the gradient of the line through the points.
Find the mid-point and perpendicular bisector of the line segment joining the points.
Enlarge the second point in relation to the first in a ratio you specify.
Create a circle centred at the first point with the second on the circumference.
Create a quadratic through the two points, with a gradient at the first point which you
specify.
Make the selected points 2 of n equally space points.
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Using the Object Menu 5
With 3 or more points selected you can…
Find a circle through the 3 points, simply find the centre of such a circle or find an
ellipse using the first two points as the foci and the final point as a point on the ellipse.
Find the centroid.
Find a polynomial of order (n-1) that passes through the points, find a polynomial of
order n that passes through the points and has a gradient which you specify at the first
selected point or find a best fit polynomial of other order.
Set up a quadrants test (simple correlation test).
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Using the Object Menu 6
With two graphs selected you can
Find the points where the curves intersect. (They are shown on the graph, and the coordinates are placed in the Results Box and can be accessed by clicking
).
Find the area between the curves.
If one of your two graphs is y = x you can perform an x = g(x) iteration.
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3D Graphs
Selecting ‘New 3D Graph Page’
dimensions.
allows you to explore graphs and vectors in 3
Equations can be entered in a variety of forms and once the graph is drawn you can
alter the viewing angle (click somewhere in the cube defined by the axes and drag, you
should get a hand shaped cursor, until you get the view you want. It can be tricky to get
just the right view.)
In addition to graphs, you can enter vector equations of lines and planes. (Unfortunately
you can’t enter Cartesian equations of lines in 3D, Autograph doesn’t like two = signs in
the same ‘equation’.).
With a point attached to a graph you can draw normal vectors and tangent planes.
With two planes selected you can find the line of intersection.
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Statistical Graphs
Selecting New 1D Statistics Page
allows you to explore single variable data.
If you have a set of data in the form of a list and you would like to draw a histogram, a
cumulative frequency graph, a box whisker diagram or a stem and leaf plot AutoGraph
will draw these for you as well as working out a range of statistics for the data.
The best approach is to select “Enter Grouped Data”
box on the next page.
which gets you the dialog
Having entered your data you can then draw the graphs you want simply by clicking on
the relevant button.
You can also produce a table of statistics, or list the relevant statistics in a box by
clicking on either
or
.
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Statistical Graphs 2
Click on Use Raw
Data, then the Edit
button
Then enter the list of
data in the next dialog
box.
Now enter the limits of
the groups you wish to
use. Click recalculate to
see how well they work.
(If you haven’t looked at
the max and min, you
can put in too many
groups and whittle them
down as required.)
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