Assignment 1

I chose to write up problem #3:

3. Find two linear functions f(x) and g(x) such that their product

is tangent to each of f(x) and g(x) at two distinct points P and Q. Discuss and illustrate the method and the results.

I began this assignment by writing down a set of equations involving f(x), g(x), and h(x) such that the above conditions were satisfied. I then commenced work on a two page mess of algebraic manipulations which led to quite a disheartening discovery involving the two supposedly distinct points P and Q mentioned above: P = Q. I had obviously made a mistake in the sea of algebra I had produced. This purely algebraic approach was probably not the approach Dr. Wilson had intended us to take. Rather than wade through the algebra to find my mistake, and since this is a course involving technology and mathematics, I thought it might be more efficient for me to experiment with these functions using Graphing Calculator 3.5.

Since slope and y-intercept uniquely define a line, I set up an environment in Graphing Calculator that would allow me to experiment with how the graphs of the three functions change as the slopes and y-intercepts of the linear functions change. The following input made it easy for me to experiment with different values of slopes and y-intercepts:

m_1=slider([-10,10])

m_2=slider([-10,10])

b_1=slider([-10,10])

b_2=slider([-10,10])

function(f_1,x)=m_1*x+b_1

function(f_2,x)=m_2*x+b_2

y=function(f_1,x)

y=function(f_2,x)

y=function(f_2,x)*function(f_1,x)

Graph of the formula
(View and edit these equations live.)

As I observed the changes in the graph made by adjusting the slopes and y-intercepts, I made several key observations:

  1. h(x) will always be a parabola.
  2. The roots of the two linear functions seemed to be the two roots of the parabola.
  3. If the slopes of the two lines were not opposite, it seemed the required conditions could not be met
  4. When the slopes were opposite, for any given value of one y-intercept, it always seemed possible to find a value of the other y-intercept so that the required conditions were met.

If I could prove that these four observations always held, the problem would be solved, and in addition to being able to find two linear functions maintaining the required conditions, I would have found a characterization of all such functions.

The proofs that these observations do always hold follow:

  1. That h(x) is a parabola is rather obvious:
  2. This follows from the simple fact that a*b = 0 when either a = 0 or b = 0. (Just substitute a with f(x) and b with g(x).)
  3. From step 2, we know that the points of tangency are where our parabla, which is of the form , is equal to zero. We can then find the slope of the tangent line at these two points by evaluating at . The two slopes are thus and , thus the slopes must be opposite of one another.
  4. Our functions can be written as follows:



    From observation #1 and #2, we know that f(x) = g(x) only where f(x) = 0.
    Since f(x) = 0 when , the only way f(x) can be tangent to g(x) is if


    and since from #3 we know that for this to happen, . Therefore
    and dividing by , which means f(x) and g(x) can not have zero slopes, we get that
    .

    We have thus characterized all pairs of linear functions f(x) and g(x) such that their product is tangent to each of f(x) and g(x) at two distinct points. They have the following form:
    and

We can check these results by inputting the following lines into Graphing Calculator, which will allow us to to graph f(x), g(x), and h(x) simultaneously, while changing the values of m and b dynamically:

m=slider([-10,10])

b=slider([-10,10])

function(f,x)=m*x+b

function(g,x)=-(m*x)+1-b

y=function(f,x)

y=function(g,x)

y=function(f,x)*function(g,x)

Graph of the formula

This file was created by Graphing Calculator 3.5.
Visit Pacific Tech to download the helper application to view and edit these equations live.

It is important to understand where the technology played its role in helping me solve this problem. Being able to dynamically change the two slopes and two y-intercepts and instantaneously see the graphs of the three functions was essential in allowing me to arrive geometrically, rather than algebraically, at statements 3 and 4 above. This geometric reasoning gave the problem an extra layer of meaning, certainly enhancing my understanding. It is also worth noting that it might have taken me quite a while to solve the problem using a purely algebraic approach. (Although this time would not be wasted time, as understanding the relationship between the geometry and algebra of a problem is quite important. It seems that in many students, there is too often a disconnect between geometry and algebra.)

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