Wavy Curve Method.
Wavy Curve method is a tool used to evaluate the inequalities of the form, where we compare the function with respect to zero.
- Values of x for which f(x) > 0.
- Values of x for which f(x) $\geq$ 0.
- Values of x for which f(x) < 0.
- Values of x for which f(x) $\leq$ 0.
Wavy Curve method work on the principle that the function f(x) = $\cfrac{p(x)}{g(x)}$ change there sign only at the zeros of f(x) or at vertical asymptotes of f(x).
We know that any continuous function whenever change its sign from positive to negative ( or negative to positive ) will become zero in between at the zero of the expression. This very fact is used to draw the wavy behaviour of the function.
- (x – 3) > 0 for x > 3 and (x – 3) < 0 for every x < 3. Hence (x – 3) Changes from negative to positive with respect to value of x where (x – 3) = 0.
- Also where ever (x- 3) > 0 the expression $\cfrac{1}{(x-3)}$ > 0. Also Where ever (x – 3) < 0 the expression $\cfrac{1}{(x-3)}$ < 0. Hence the function f(x) = $\cfrac{1}{(x-3)}$ changes its sign with respect to the vertical asymptote of f(x).
Drawing of Wavy Curve Behaviour.
When none of zero or vertical asymptote is repeated.
For any function of form f(x) = $\cfrac{p(x)}{g(x)}$ the following steps are performed.
e.g Let f(x) =$\cfrac{x^2-x}{x^2+x-6}$
STEP 1 :: Factorize Numerator and denominator into linear factors.
i.e f(x) =$\cfrac{x(x-1)}{(x+3)(x-2)}$
STEP 2 :: Draw the real number line. Mark the points, which are either zeros of f(x) and where there exist vertical asymptotes to f(x). ( We can name these points as nodal points ). The nodal points where there exist vertical asymptotes, are to be always marked as open interval point.
i.e Mark points where x(x-1) =0 and where (x+3)(x-2)=0. These are x=-3, 0, 1, 2 where vertical asymptotes are at x=-3 and x=2 Hence mark open interval.
STEP 3 :: Check is f(x) is positive or negative for the value of x greater than the rightmost nodal point ( x greater than the greatest nodal point ) marked on real number line. Now starting with this point draw wave crossing real axis at nodal point only.
put x=3 in f(x) , i.e f(3) = $\cfrac{3(3-1)}{(3+3)(3-2)}\gt 0$
The curve obtained above is the wavy behaviour of f(x)=$\cfrac{x^2-x}{x^2+x-6}$.
The wavy curve indicate interval of x for which f(x) is greater than zero and interval values of x where f(x) is less than zero.
i.e for f(x)=$\cfrac{x^2-x}{x^2+x-6}$
Where f(x) > 0 for $x\in (-\infty, -3)\cup (0,1)\cup (2, \infty)$
f(x) < 0 for $x\in (-3, 0)\cup (1, 2)$.
When zero or vertical asymptote is repeated.
For any function of form f(x) = $\cfrac{p(x)}{g(x)}$, given that zero’s of f(x) or vertical asymptotes or both are repeated.
e.g Let f(x) =$\cfrac{x^3-x^2}{x^2+x-6}$
STEP 1 :: Factorize Numerator and denominator into linear factors.
i.e f(x) =$\cfrac{x^2(x-1)}{(x+3)(x-2)}$
STEP 2 :: Draw the real number line. Mark the points, which are either zeros of f(x) and where there exist vertical asymptotes to f(x). ( We can name these points as nodal points ). The nodal points where there exist vertical asymptotes, are to be always marked as open interval point.
i.e Mark points where $x^2(x-1) =0$ and where (x+3)(x-2)=0. These are x=-3, 0, 1, 2 of which x=0 is repeated zero of f(x). Where vertical asymptotes are at x=-3 and x=2 Hence mark open interval.
STEP 3 :: Check is f(x) is positive or negative for the value of x greater than the rightmost nodal point ( x greater than the greatest nodal point ) marked on real number line. Now starting with this point draw wave crossing real axis at nodal point only.
put x=3 in f(x) , i.e f(3) = $\cfrac{3^2(3-1)}{(3+3)(3-2)}\gt 0$
The curve obtained above is the wavy behaviour of f(x)=$\cfrac{x^3-x^2}{x^2+x-6}$.
i.e for f(x)=$\cfrac{x^3-x^2}{x^2+x-6}$
Where f(x) > 0 for $x\in (-3, 0)\cup (0,1)\cup (2, \infty)$
f(x) < 0 for $x\in (-\infty, -3)\cup (1, 2)$.