The purpose of this lab is to become more familiar with Maple by using it to investigate manipulations of linear and quadratic functions.
The first part of this lab deals with linear functions, that is,
functions of the form 
, where a and b are the slope
and intercept. In Maple, it is easy to define both particular linear
functions ( a and b are numbers) and general linear functions (a
and b are parameters), as shown in the examples below.
> f := x -> x+2 ;
> g := x -> -2*x-3 ;
> f(g(x));
> h := x -> a*x+b;
> h(x);
The third example above shows how to form the composite function
. The last example shows how to define a general linear
function, where a and b are parameters.
In the exercises, you will need to know when two linear functions, say
and 
are
equal. This means that the equation 
is true for
any value of x. ( More generally, two arbitrary functions 
and 
are equal if the equation 
is true for
any value of x.)
Geometrically, this means that the graphs of the two lines
coincide. Since the equation of a linear function is determined by its
graph, we can also say that two linear functions are equal if they
have the same slopes and intercepts. For example, the two linear
functions 
and 
we defined in the examples above are not equal, because
their slopes and intercepts are not the same.
Now, suppose we wanted to choose values of a and b so that 
was equal to 
. Using the fact that the slopes and intercepts
have to be the same, we see that we would have to choose a=1 and b=2.
and 
. Form the composite
function 
and verify that the resulting
function is also
linear. Is this always the case? Make a conjecture and use Maple to
help you prove it! That is, if 
and 
, is
it always true that 
is a linear function?
and 
. Is
it always possible to find another linear function 
so that
?
Try this out with the following pair of functions. Then make a general
conjecture and prove it. In your writeup, make sure that you state
clearly what your conjecture is and also describe how you prove it.
has the form
. Suppose that 
. What conditions on
the constants 
are necessary for the
two quadratic functions 
and 
to be equal?