Eric Gold

Assignment 1, Part 3:

Tangential Linear Equations

Find two linear functions f(x) and g(x) such that their product, h(x) = f(x)*g(x), is tangent
to each of f(x) and g(x) at two distinct points.

Consider the simplest case.
For the both lines to be tangential to the curve, they must have
negative reciprocal slopes. This
allows the vertex of the parabola and the intersection of the lines to be
collinear. That is to say, the
vertical line through the intersection of the 2 lines is the axis of symmetry
of the parabola. The simplest such
lines are f(x) = x and g(x) = -x:

**Graph A**

Taking
the product gives me h(x) = -x^{2,}, in Graph
B.

**Graph B**

Note
that either the linear functions or the parabola need to be shifted in order
for *f * and *g* to be
tangential to *h*. It makes sense to try and raise the
linear graphs above the parabola by changing the y-intercepts, first raising
them by 1 as follows:

f(x) = x + 1

g(x) = -x + 1

h(x) = (x + 1)(-x + 1)

By
observation of Graph C, this is too much, so consider splitting the difference
between the 2 intercepts, lowering them by ½:

**Graph C**

This
results in Graph D.

g(x) = -x + ½

h(x) = (x + ½)(-x + ½)

Note that the curve *h* appears to
be tangent to the functions *f* and *g*. The vertical shifts
in the graphs have maneuvered the lines in such as way that have left us with
the linear functions apparently tangent to the graph of the parabola that is
the product of them at the roots of the parabola as seen in Graph D,
immediately below.

**Graph
D**

This can be verified by increasing the
scale and looking more closely at the closed interval for x [
-1, 1]:

**Graph
E**

Observe:

By manipulating the product of a sum and a difference, we were
able to manipulate the linear functions up and down the y-axis until we reached
the desired condition, specifically that the linear functions both be
tangential to the function that is their product.

We could continue to work with the function and manipulate it in
such ways that we shift the graph horizontally, that we stretch or compress the
graph, *i.e.*
alter how fast the parabola changes, or any combination of all three features
of the graph.