Two Linear Functions Each Tangent to The Product Function:
The proposed investigation is for finding two linear functions
f(x) = ax + b
g(x) = cx + d
such that the graphs of each of these lines is tangent to the product function
h(x) = f(x).g(x)
The ususal approach is to do the algebra to get the product
Then various manipulations with algebra or even calculus are used to eventually find something like the graph at the right.
The alternative used in this note is to begin with the linear equations
y = ax + b
y = cx + d
and write them in general form
ax - y + b = 0
cx - y + d = 0
These produce the same linear graphs as the y = mx + b format. The product graph
(ax - y + b)(cx - y + d) = 0
would generate all the points on either of the lines since if (x,y) is on either of the lines then it also a point on the product. For instance the graph at the right shows
(2x - y -1)(x - y +1) = 0
Now, examine the expanded product:
Now, the first three terms are the desired h(x) product function. If the xy and coefficients are 0 and the y coefficient is -1, then an equation for y = h(x) would result. So let
a + c = 0 and b + d = 1
Set the coefficient of equal to 0. Here is the graph when a = 2, c = -2, b = 0 and d = 1.
The a + c = 0 and b + d = 1 are the key parameters. From this it is now possible to observe and prove
i. The tangent points are on the x-axis. By the product, the points where the graphs of the linear functions cross the x-axis must be on the graph of the product function as well; the product function must have roots at the same points as the two lines. Further, if there were other points where the graphs crossed, then the combined graph would be a factorable expression. That can happen only when the coefficient of the term is 1. In that case we have the degenerate case of the combined graph being the two intersecting lines.
The product function will be a quadratic. When the coefficient of is 0 we know the curve in a parabola.
ii. When f(x) = 0, g(x) = 1 and when g(x) = 0, f(x) = 1.
iii. The y-coordinate for the intersection of the linear functions is
Set c = -a and d = 1 - b. To find (x,y) that is the point of intersection set
ax + b = -ax +1 -b
2ax = 1 - 2b
The x-coordinate for this point of intersection can also be expressed in terms of c and d and it identifies the axis of the parabola.
iv. The y-coordinate for the vertex of the product function is
In the expanded product quadratic, when c = -a, the xy term disappears, thus guaranteeing that the resulting quadratic will be symmetric about a vertical axis, For each value of a coefficient of the term other than 1, the product is a conic tangent to each of the lines.
If the coefficient is 1, the graph is the pair of lines.
In the coefficient is ac, the graph is a circle.
If the coefficient is negative (other than ac) the graph is an ellipse.
If the coefficient is between 0 and 1 the graph in a hyperbola within the vertical angles of the two lines.
If the coefficient is greater than 1 the graph is a hyperbola within the horizontal angles of the two lines.
Here is a movie of changes in the product graph when c = -a, d = 1 - b, and the coefficient of being varied.