The Gateway Arch as a quartic function

As a scientist or engineer might tell you, polynomials can be indispensable in creating or approximating any number of smooth shapes. For example, if you plot the polynomial equation y = -3.87×10-8x4 - 0.00346x2 + 625, its graph matches very closely the shape of the Gateway Arch in St. Louis (with x and y in feet).

Because of the shape of the Arch (moving downward to the left and to the right) it makes sense that an even-degree polynomial with a negative leading coefficient should be used to describe that shape. But as it turns out, the graph of a second-degree polynomial (a quadratic function whose graph is a parabola) would be noticeably "pointier" than the Arch.

In reality, the exact shape of the Arch comes from a more advanced function than those that we deal with in an algebra course, but the polynomial above is close enough for many practical purposes. For example, suppose we wish to hang a banner from the Arch, 500 feet above the ground. How wide should it be?

The following graph shows the fourth-degree polynomial equation given before, along with the five actual Arch measurement points used to create it. Try moving these points around to create graphs of other fourth-degree polynomial equations. Try switching to a second-degree polynomial and see how well you can make it fit the shape of the Arch.

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