Vertical curve equation – upgrade g1% followed by downgrade g2% For a parabolic vertical curve of length L (measured horizontally), joining an initial grade of g1% to a final grade of g2%, the elevation y at a distance x from the point of curvature (PVC) is expressed as:

Introduction / Context:
Highway vertical curves are designed as simple parabolas so that the gradient changes at a constant rate, giving comfort and adequate sight distance. For design and setting out, a compact elevation equation is needed.

Given Data / Assumptions:

• Initial grade = g1% (upgrade positive).
• Final grade = g2% (downgrade may be negative relative to g1).
• Curve length = L (horizontal).
• Origin x = 0 at PVC; y measured from the tangent at PVC.

Concept / Approach:
For a parabola y = a x^2 + b x, slope dy/dx = 2 a x + b. At x = 0, slope must equal initial grade in fraction g1/100 → b = g1/100. At x = L, slope must equal final grade g2/100 → g2/100 = 2 a L + g1/100, giving a = (g2 − g1) / (200 L).

Step-by-Step Solution:

Verification / Alternative check:
Second derivative d^2y/dx^2 = 2 a = (g2 − g1)/(100 L), which is the constant rate of change of grade—exactly as required for a parabolic vertical curve.

Why Other Options Are Wrong:
Options B and D reverse start/end slopes or use an incorrect denominator. Option C adds grades, violating boundary conditions. Option E is linear, not parabolic.

Common Pitfalls:
Mistaking percent grade for fractional slope; mixing up PVC/ PVT reference; forgetting the factor 200 when grades are in percent.

Final Answer:
y = (g1/100) * x + ((g2 - g1) / (200 * L)) * x^2