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1. If in an equation of nth degree, the term with xn is absent, then the coefficient of the next higher power of x when equated to zero, gives horizontal asymptotes (provided it must not be a constant)

For example, x3 + 3xy2 + y2 + 2x + y = 0 has no horizontal asymptote as x3 is present

For example x2 y3 + x33y2 = x3 + y3

This equation is a 5th degree equation. Since x5 is absent, we have coefficient of x3 is (y2 - 1)= 0 i.e. y = ± 1 are two asymptotes parallel to x-axis i.e. horizontal asymptotes.

Note that the ‘Oblique asymptotes’ concept is beyond the scope of this book.

Example 1 Sketch the graph of the function x2 + 2x - 5 (1) using it determine the real roots of x2 + 2x - 5 = 0 (2) Find y when x = 2.5

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Solution :

1) Let y = f(x) = x2 + 2x - 5

 x - 4 -3 -2 -1 0 1 2 3 y = f(x) 3 -2 -5 -6 -5 -2 3 10

The curve is a parabola which cuts x-axis in points whose abscissa is between 1 and 2 and a point whose abscissa is between -3 and 4. From the graph the roots are x =1.5 and x = -3.5. From the graph y = 6.25 when x = 2.5.

Index

Introduction

1.1 Functions And Mapping
1.2 Functions, Their Graphs and Classification
1.3 Rules for Drawing the Graph of a Curve
1.4 Classification of Functions
1.5 Standard Forms for the equation of a straight line
1.6 Circular Function and Trigonometry

Chapter 2