2.42. Explain why the second differences of curve elevations are equal for a parabolic curve.

Try these first

1. 1.

   Define a parabolic curve.

   Hint

   Remember the equation form y = ax² + bx + c.

2. 2.

   What are second differences in terms of elevation?

   Hint

   Think about how you find the differences in a series of values.

3. 3.

   What type of equation defines a parabolic curve?

   • Linear
   • Quadratic
   • Cubic

   Hint

   Recall the form y = ax² + bx + c.

4. 4.

   Are the second differences of a parabolic curve equal?

   • True
   • False

   Hint

   Think about the mathematical characteristics of parabolas.

5. 5.

   Given a parabolic equation y = 2x² + 3x + 4, calculate the second differences for x = 0, 1, 2, and 3.

   Hint

   Focus on calculating successive differences at equal intervals.

6. 6.

   If a road is designed with a parabolic vertical curve connecting a 5% to a 1% gradient, what elevation modeling will ensure second differences remain equal?

   Hint

   Map how gradients could be approached using quadratic functions.