## 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 $x\in (-\infty, -3)\cup (1, 2)$.