Engineers typically use a polynomial to describe the shape of vertical curves. The equation of
three vertical curves is given. Each represents the shape of the road for a distance of x feet
for {x|0 x 400}.
• Vertical Curve A: y = –0.000 112 5x
2 + 0.04x + 100
• Vertical Curve B: y = 0.000 112 5x
2 + 0.02x + 100
• Vertical Curve C: y = −0.0003x
2 + 0.12x + 100
a. What type of polynomial is represented by each equation?
b. By just looking at the equations, which of the curves do you expect to represent similar
driving conditions? Explain your choice.
c. Describe the end behaviour you would expect to see for a polynomial used to make a
road over a hill and the end behaviour you would expect to see used to make a road
through a valley
a. Vertical Curve A and Vertical Curve C represent second-degree polynomials, while Vertical Curve B represents a second-degree polynomial as well.
b. Based on the coefficients of the linear and quadratic terms in the equations, Vertical Curve A and Vertical Curve C have the same quadratic term but different linear terms, while Vertical Curve B has the same linear term as Vertical Curve A but a different quadratic term. Therefore, Vertical Curve A and Vertical Curve C are expected to represent similar driving conditions, with a steeper slope compared to Vertical Curve B.
c. For a polynomial used to make a road over a hill, the end behavior would typically involve the polynomial increasing on one end and decreasing on the other end. This would result in a point where the curve reaches its highest point before descending. On the other hand, for a polynomial used to make a road through a valley, the end behavior would involve the polynomial decreasing on both ends, forming a U-shaped curve to represent the valley.