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Linear and Quadratic Functions

Purpose

The purpose of this lab is to become more familiar with Maple by using it to investigate manipulations of linear and quadratic functions.

Background

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.

Exercises

  1. Suppose 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?

  2. Suppose you are given two linear functions 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.

  3. A general quadratic function has the form . Suppose that . What conditions on the constants are necessary for the two quadratic functions and to be equal?



next up previous
Next: Limits of functions. Up: Labs and Projects for Previous: Maple Introduction Exercises



William W. Farr
Mon Jun 26 13:37:30 EDT 1995