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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.

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.

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