Linear Functions
by Morgan Guest
Finding two linear functions f(x)=ax+b and g(x)=cx+d, a,c ≠0 such that their product h(x) =f(x).g(x) is tangent to each of f(x) and g(x) at two distinct points.
, f(x) and g(x) are linear expressions. Call
and
So,
.
Example: If h(x) is tangent to f(x) and g(x) only once then h(x) = f(x) at one point and h(x) = g(x) at one point.
When does f(x) intersect h(x)?
Either ax + b = 0 or cx + d – 1 = 0. Therefore,
or
. There can only be one tangent value so
à
When does g(x) intersect h(x)?
Either cx + d = 0 or ax + b – 1 = 0. Therefore,
or
à
By plugging back into the equation
, you find that
à
In terms of a and b,
,
,
Simplify h(x):
We want to find where f(x) and g(x) intersect.
Intersection point:
. This mathematically shows that the lines intersect at the same height no matter their slope or where it crosses y axis.
As we have seen before, the tangent points must occur at
and
. Both of these points are on the x axis where y=0.
We want to find the vertex:
Vertex: (
)
The height of the vertex is always at y= 1/4 and is not dependent on the values of a, b, c, or d.
