Sag vertical curves Refer to Exercise. Valleys or dips in highways are referred to as sag...

Sag vertical curves Refer to Exercise. Valleys or dips in highways are referred to as sag vertical curves. Sag vertical curves are also modeled using parabolas. Two roadways with different grades meeting at a sag curve need to be connected. The highway passes through the points ,B(0, 110), C(750, 10), D1500, 110), and , as shown in the figure. The roadway is linear between A and B, parabolic between B and D, and linear between D and E.

Crest vertical curves When engineers plan highways, they must design hills so as to ensure proper vision for drivers. Hills are referred to as crest vertical curves. Crest vertical curves change the slope of a highway. Engineers use a parabolic shape for a highway hill, with the vertex located at the top of the crest. Two roadways with different slopes are to be connected with a parabolic crest curve. The highway passes through the points A(−800, −48), B(−500, 0), C(0, 40), D(500, 0), and E(800, −48), as shown in the figure. The roadway is linear between A and B, parabolic between B and D, and then linear between D and E.

EXERCISE

(a) Find a piecewise-defined function f that models the roadway between the points A and E.

(b) Graph f in the viewing rectangle [−800, 800, 100] by [ −100, 200, 100].

EXERCISE

(a) Find a piecewise-defined function f that models the roadway between the points A and E.

(b) Graph f in the viewing rectangle [−500, 2000, 500] by [0, 800, 100].