Sign Analysis on the Real Number Line: Linear and Quadratic Models
Determining the sign of a mathematical function is a cornerstone of functional analysis and optimization. Investigating the sign of a function f(x) entails discovering the intervals of the real number line ℝ on which the function values are strictly positive (f(x) > 0), strictly negative (f(x) < 0), or zero (f(x) = 0). In Secondary 1, learners master the sign analysis of constant functions, linear functions f(x) = ax + b, and quadratic functions f(x) = ax² + bx + c. The sign of a quadratic function is intimately governed by the discriminant Δ and the orientation of the parabola (the sign of the leading coefficient a). This conceptual foundation provides the requisite tools for solving non-linear inequalities.
1. Sign of Constant and Linear Functions
Constant Function: f(x) = c (c ≠ 0)
The function has the same sign as c for all real numbers x ∈ ℝ.
Example: f(x) = 5 is strictly positive on ℝ. f(x) = -3 is strictly negative on ℝ.
Linear Function: f(x) = ax + b (a ≠ 0)
1. Find zero: ax + b = 0 ⟹ x = -b/a.
2. On the number line:
• For x > -b/a (to the right): Same sign as a.
• For x < -b/a (to the left): Opposite sign to a.
2. Sign of Quadratic Function: f(x) = ax² + bx + c (a ≠ 0)
The sign distribution depends entirely on the discriminant Δ = b² - 4ac:
| Discriminant | Roots of f(x) = 0 | Sign Distribution across the Real Number Line |
|---|---|---|
| Δ < 0 | No real roots | f(x) has the SAME sign as 'a' for ALL x ∈ ℝ. It never changes sign. |
| Δ = 0 | One repeated real root x = -b/(2a) | f(x) = 0 at x = -b/(2a). For all other x ∈ ℝ \ {-b/(2a)}, f(x) has the SAME sign as 'a'. |
| Δ > 0 | Two distinct real roots x₁ < x₂ |
• For x ∈ (x₁, x₂) (between roots): OPPOSITE sign to 'a'. • For x ∈ ℝ \ [x₁, x₂] (outside roots): SAME sign as 'a'. • At x ∈ {x₁, x₂}: f(x) = 0. |
Sign Investigation Workshop
Investigation: Sign of f(x) = -x² + 4x - 3
Determine where f(x) is positive, zero, and negative.
1. Leading coefficient: a = -1 (negative).
2. Solve -x² + 4x - 3 = 0 ⟹ x² - 4x + 3 = 0 ⟹ (x - 1)(x - 3) = 0 ⟹ Roots: x₁ = 1, x₂ = 3.
3. Sign analysis:
• Between the roots x ∈ (1, 3): Sign is opposite to 'a' ⟹ f(x) > 0 (Positive).
• At x ∈ {1, 3}: f(x) = 0.
• Outside roots x ∈ ℝ \ [1, 3]: Sign is same as 'a' ⟹ f(x) < 0 (Negative).
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