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