Wavy Curve Method or Sign Scheme for Rational Functions

1. Concept Overview

The wavy curve method (also called the sign scheme method) is a systematic way of determining where a rational expression is positive, negative, or zero.

A rational function is of the form:

[ f(x) = \dfrac{N(x)}{D(x)} ]

where both numerator and denominator are polynomials.

To analyze the sign of the rational function, we:

1. Factor the numerator and denominator into linear factors.
2. Mark the critical points (roots of numerator and denominator) on the number line.
3. Use a wavy curve to check the sign in each interval.
4. Identify where the expression is positive, negative, or zero.

2. Why We Use the Wavy Curve Method

For rational inequalities, manual sign checking for each interval is time-consuming.
Instead, the wavy curve method visually shows:

• where the curve crosses the x-axis
• points where it changes signs
• where the expression becomes undefined
• which intervals satisfy a given inequality

3. Detailed Steps

Step 1 — Factor Completely

Write:

[f(x) = \dfrac{N(x)}{D(x)} ]

Factor both numerator and denominator into linear factors:

[N(x) = a(x – \alpha_1)(x – \alpha_2)\ldots ]
[D(x) = b(x – \beta_1)(x – \beta_2)\ldots ]

Step 2 — Mark Critical Points

Critical points include:

• zeros of numerator (where expression becomes 0)
• zeros of denominator (where expression is undefined)

Place them on a number line in ascending order.

Step 3 — Draw the Wavy Curve

Between consecutive critical points, alternate the sign (+ – + – + …).

Important rule:

If the total exponent of a repeated factor is even, the sign does NOT change at that point.

Step 4 — Determine Positive/Negative Intervals

Using the wavy curve:

• above the line → positive
• below the line → negative

4. Example (with explanation)

Example 1
Analyze the sign of:

[ f(x) = \dfrac{x^3 – 4x}{x^2 – 4} ]

Step 1: Factor

Numerator:
[ x^3 – 4x = x(x^2 – 4) = x(x – 2)(x + 2) ]

Denominator:
[ x^2 – 4 = (x – 2)(x + 2) ]

Thus,

[ f(x) = \dfrac{x(x-2)(x+2)}{(x-2)(x+2)} ]

Cancel only for sign purposes (NOT for domain):
Effective expression:
[ x ]

Critical points:
• [x = -2] (denominator zero)
• [x = 0] (numerator zero)
• [x = 2] (denominator zero)

Step 2: Number line

Mark points −2, 0, 2.

Step 3: Determine signs

Check sign in each interval using wavy curve:

Final answer:

[f(x) > 0] for [(−2, 0) ∪ (2, ∞)]
[f(x) < 0] for [(−∞, −2) ∪ (0, 2)]

5. More Examples

Example 2
[ f(x)=\dfrac{x+3}{x-1} ]

Critical points: [x = −3] (zero), [x = 1] (undefined)

Intervals:
[(−∞, −3)]: negative
[(−3, 1)]: positive
[(1, ∞)]: positive

Example 3
[ f(x)][=\dfrac{(x-4)(x+1)}{(x-2)} ]

Critical points: [x = −1, 2, 4]
Signs alternate unless multiple roots exist.

6. Important Notes

• Undefined points are always excluded from the solution set.
• Zeros of numerator are included only if inequality uses ≥ or ≤.
• Repeated roots require special attention:
– even multiplicity → sign does NOT change
– odd multiplicity → sign changes