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 is represented by a quadratic polynomial, Vertical Curve B is represented by a quadratic polynomial, and Vertical Curve C is represented by a 2nd degree polynomial.
b. Vertical Curve A and Vertical Curve B are expected to represent similar driving conditions as they both have positive coefficients for the x^2 term which implies a concave upward shape for the curve. Both curves also have similar coefficients for the x term, which means they will have a similar slope along the curve.
c. For a polynomial used to make a road over a hill, you would expect to see that the end behavior is increasing as x approaches positive infinity, and decreasing as x approaches negative infinity. This is because the road is going up a hill. Conversely, for a polynomial used to make a road through a valley, you would expect to see that the end behavior is decreasing as x approaches positive infinity, and increasing as x approaches negative infinity. This is because the road is going through a valley.