Additional Mathematics – Quadratic functions | e-Consult

Substituting y = mx + c into the quadratic gives:

x² − 4x + 3 = mx + c → x² + (−4 − m)x + (3 − c) = 0.

The discriminant of this quadratic in x is:

Δ = (−4 − m)² − 4·1·(3 − c) = (m + 4)² − 4(3 − c).

Hence:

• Two distinct intersections: Δ > 0 → (m + 4)² > 4(3 − c).
• Tangency: Δ = 0 → (m + 4)² = 4(3 − c).
• No intersection: Δ 

These inequalities give the required relationship between m and c. For example, solving the equality for tangency yields:

c = 3 − ½(m + 4)².