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.

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.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic |
---|---|---|---|

AP Calculus AB (USA) | 2 | Coordinate geometry | Rational curves |

AP Calculus BC (USA) | 2 | Coordinate geometry | Rational curves |

AQA A-Level (UK - Pre-2017) | FP1 | Coordinate geometry | Rational curves |

AQA A2 Maths 2017 | Pure Maths | Extra | Rational Curves |

AQA AS/A2 Maths 2017 | Pure Maths | Extra | Rational Curves |

CCEA A-Level (NI) | C3 | Coordinate geometry | Rational curves |

C Coordinate Geometry in (x,y) | Extra | Rational Curves | |

Edexcel A-Level (UK - Pre-2017) | C3 | Coordinate geometry | Rational curves |

Edexcel A2 Maths 2017 | Pure Maths | Extra | Rational Curves |

Edexcel AS/A2 Maths 2017 | Pure Maths | Extra | Rational Curves |

I.B. Higher Level | 2 | Coordinate geometry | Rational curves |

I.B. (MSSL) | 4 | Coordinate geometry | Rational curves |

OCR A-Level (UK - Pre-2017) | C4 | Coordinate geometry | Rational curves |

OCR A2 Maths 2017 | Pure Maths | Extra | Rational Curves |

OCR MEI A2 Maths 2017 | Pure Maths | Extra | Rational Curves |

OCR-MEI A-Level (UK - Pre-2017) | C4 | Coordinate geometry | Rational curves |

Universal (all site questions) | C | Coordinate geometry | Rational curves |

WJEC A-Level (Wales) | C4 | Coordinate geometry | Rational curves |