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1.3 Rules For Drawing The Graph Of A Curve (representing a function) :

  • Symmetry :

    1. If x is replaced by -x but the equation remains the same, the graph shall be symmetrical about y-axis.

    2. If y is replaced by - y but the equation remains the same, the graph shall be symmetrical about x-axis.

      For example : y2 = 4ax is symmetrical about x-axis.

      x2 + y2 = a2 is symmetrical about both x and y axis.

    1. If both x and y are changed by - x and - y respectively and simultaneously,and equation is unchanged, then the graph is symmetrical about the origin and symmetrical in opposite quadrants .

      In above illustration, x2 + y2 = a2 is a standard circle having center at the origin with radius ‘a’ units.

    2. When x and y are interchanged and equation remains the same, the curve is symmetrical about the line y = x.

      Clearly the curve x3 + y3 = 3axy is symmetrical about y = x.

  • Intercepts on the axes : Put y = 0 in the equation to find x-intercept and put x = 0 to find y-intercept.

  • Extent of the curve : Know the domain of the curve that exists. This gives the horizontal extent. Know the range, this gives the vertical extent of the curve (i.e. function)

  • Tangent at the origin : If the curve passes through origin, then equate its lowest degree term to zero. This gives the tangents at the origin.

    For example, y2 = 4ax, passes through origin

    \ Taking 4ax = 0 gives x = 0 ( i.e. y-axis )

    Similarly x2 = 4by, passes through origin

    \ Taking 4by = 0 gives y = 0 ( i.e. x - axis )

    The curve x3 + y3 = 3axy also passes through origin.

    \ Taking 3axy = 0 or x = 0 and y = 0 i.e. both axes touch the curve at the origin.


  • Asymptotes : The value of x which makes y infinitely large gives a straight line which touches a branch of the curve at infinity. Similarly for y.

    Such a straight line is called an Asymptote to the curve.

    1. To locate vertical asymptotes (parallel to y-axis), if an equation of nth degree, the term with yn is absent, then the coefficient of the next highest power of y when equated to zero, gives the vertical asymptotes (provided this coefficient must not be a constant.

      For example, x3+ 3xy2 + y2 + 2x + y = 0

      Since y3 is absent. The coefficient of y2 is ( 3x + 1 ) = 0 which the asymptote parallel to y-axis i.e. vertical asymptote.

     

  • 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





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