Example 3

Example 3

An amusement park's roller coaster has a 'hill' with a parabolic shape. The hill begins and ends at the same height, and the horizontal distance between the beginning and end of the hill is 122 m. It reaches a height of 39 m after a horizontal distance of 30 m. Determine the equation of a quadratic function that represents the height of the hill as a function of the horizontal distance. Set (0, 0) as the beginning of the hill.

Step 1: Organize the information that is given.

Sketch a graph of the known information.

The beginning of the hill is at (0, 0). Since the horizontal distance between the beginning and end of the hill is 122 m, the hill should end at (122, 0). It was also given that after a horizontal distance of 30 m, the hill is 39 m high, so (30, 39) is another point on the graph of the function.

Step 2: Select a form that works with the given information.

Because the x-intercepts are known, the factored form would be a good form to work with. The x-intercepts are 0 and 122, so two factors are (x − 0) and (x − 122) . However, it is possible that there is also a GCF. As such, the function can be stated as

or

, where a corresponds to the GCF.

Step 3: Determine the value of a.

Substitute the known point (30, 39), into the equation of the function and solve for a.

Step 4: Final statement.

The function represents the height of the hill, h(x), as a function of the horizontal distance, x.

The function is not a model for the entire roller coaster, but rather just for the hill itself. As such, the domain must be limited to the horizontal distance that spans the hill.

Domain: .

A complete function model of the hill is , .