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




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.