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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?

## Summary/Background

The red lines will vanish when a2cd+b2-ab(c+d) < 0. For example if you keep a=3, c=1, d=2, then b2-9b +18< 0 or (b-6)(b-3) < 0, which happens when b is between 3 and 6.

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.

## 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.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopic
AP Calculus AB (USA)2Coordinate geometryRational curves
AP Calculus BC (USA)2Coordinate geometryRational curves
AQA A-Level (UK - Pre-2017)FP1Coordinate geometryRational curves
AQA A2 Maths 2017Pure MathsExtraRational Curves
AQA AS/A2 Maths 2017Pure MathsExtraRational Curves
CCEA A-Level (NI)C3Coordinate geometryRational curves
C Coordinate Geometry in (x,y)ExtraRational Curves
Edexcel A-Level (UK - Pre-2017)C3Coordinate geometryRational curves
Edexcel A2 Maths 2017Pure MathsExtraRational Curves
Edexcel AS/A2 Maths 2017Pure MathsExtraRational Curves
I.B. Higher Level2Coordinate geometryRational curves
I.B. (MSSL)4Coordinate geometryRational curves
OCR A-Level (UK - Pre-2017)C4Coordinate geometryRational curves
OCR A2 Maths 2017Pure MathsExtraRational Curves
OCR MEI A2 Maths 2017Pure MathsExtraRational Curves
OCR-MEI A-Level (UK - Pre-2017)C4Coordinate geometryRational curves
Universal (all site questions)CCoordinate geometryRational curves
WJEC A-Level (Wales)C4Coordinate geometryRational curves