The graph of $y = {(ax - b) \over (x - c)(x - d)}$

The display initially shows the above curve for $a = 3$, $b = 7$, $c = 1$, $d = 2$

Also shown are two red horizontal lines. The curve does not lie between these lines. In other words, y cannot take values between those indicated by the red lines, so y (with initial coefficients) cannot take values between 1 and 9.

What values of $a$, $b$, $c$, $d$, make the red lines vanish?

The display initially shows the above curve for $a = 3$, $b = 7$, $c = 1$, $d = 2$

Also shown are two red horizontal lines. The curve does not lie between these lines. In other words, y cannot take values between those indicated by the red lines, so y (with initial coefficients) cannot take values between 1 and 9.

What values of $a$, $b$, $c$, $d$, make the red lines vanish?

## Summary/Background

The red lines will vanish when
a

^{2}cd+b^{2}-ab(c+d) < 0. For example if you keep a=3, c=1, d=2, then b^{2}-9b +18< 0 or (b-6)(b-3) < 0, which happens when b is between 3 and 6.## Software/Applets used on this page

This page uses JSXGraph
JSXGraph is a cross-browser library for interactive geometry, function plotting, charting, and data visualization in a web browser. It is implemented completely in JavaScript, does not rely on any other library. It uses SVG and VML and is fully HTML5 compliant.
This page also uses the MathJax system for displaying maths symbols.

## Glossary

### function

A rule that connects one value in one set with one and only one value in another set.

### graph

A diagram showing a relationship between two variables.

The diagram shows a vertical y axis and a horizontal x axis.

The diagram shows a vertical y axis and a horizontal x axis.