Consider the general equation of a reciprocal function.

Reciprocal Function y = \frac{a}{x - h} + k Constant of Variation : a Horizontal Asymptote : k Vertical Asymptote : h

Let's start by graphing the parent function to all reciprocal functions,

y = \frac{1}{x} .

This function has a constant of variation of

1,

and a horizontal and vertical asymptote of

0 .

Notice that the minimum distance from the origin to this function is a $4 5^{\circ}$ line from $(0, 0)$ to $(1, 1)$ or $(- 1, - 1) .$ Let's add this to the graph.

If we zoom in on one of the quadrants, we notice that the distance is the hypotenuse of a $4 5^{\circ} - 4 5^{\circ} - 9 0^{\circ}$ triangle. In such a triangle the hypotenuse is always $2$ longer than any of the legs. Since the legs are both $1$ unit long, the hypotenuse must be $2$ units.

As we can see, when the constant of variation is $1$ the function passes through $(1, 1) .$ If we can push this graph outward so that the closest point to the origin is $(4, 4),$ we will have a reciprocal function with a minimum distance of $42$ units. This is because this point creates a $4 5^{\circ} - 4 5^{\circ} - 9 0^{\circ}$ triangle with a leg length of $4 .$

Exercise