The first section of your track will move the train horizontally from the boarding station to the beginning of the hill. The second section of the coaster will transition the train from a horizontal position to a position where it climbs the main hill of the roller coaster. The third section of your roller coaster will be the initial hill of the roller coaster. This will be the tallest hill of the roller coaster and will be in the shape of a parabola. The fourth and final section of your design will start on the downward slope of the first hill and move the roller coaster through additional dips and peaks. It will also dip below the ground into and out of a tunnel at some point.

The roller coaster leaves the boarding station and travels a horizontal distance of 10 feet on a track that is 8 feet above the ground. This function is represented by:

h(x)=8 if 0≤x<10.

The second portion of track moves the train from the horizontal position to a position where it is climbing the first hill.

This section starts where your first section leaves off. The shape of this section is half of a parabola with the vertex as a minimum. This section of track starts at the vertex and ends at a point that is 20 feet above the ground and at a horizontal position of 20 feet from that of the boarding station. The function can be represented by:

h(x)=3/25x²-12/5x+20 if 10≤x<20

For the third section: This part of the piecewise function h(x) should start at a location that is at a horizontal position of 20 feet from the boarding station and 20 feet above the ground. The movement up the hill should not have any sharp corners as you move from the second part of the track to the third part of the track like a real roller coaster would.

You are only in charge of a section of this that starts from the boarding station and ends at a horizontal position of 300 feet from the boarding station. Your hill needs to reach a maximum point and descend leaving you with enough room for the fourth section which is to have additional dips and peaks, as well as entering and exiting a tunnel at least once. Design the main hill of your roller coaster by deciding how tall and wide you want it to be. Keep in mind that the tallest roller coaster in the world was built in 2005 and is 456 feet tall. Write the function that continues h(x) meeting the requirements and pick a good endpoint to continue with the fourth function.

For the fourth section: This design must have at least one dip (relative minimum) and one peak (relative maximum). Any peak you have in section 4 must be lower than the height of the first hill. Your first hill should still be the tallest of your roller coaster. This section must also have the train dip into and emerge from an underground tunnel(x at 0) at least once. Due to many factors, the tunnel should not be too deep. For reference, the deepest tunnel in the United States is 90 feet below ground. When your coaster reaches a horizontal position of 300 feet from that of the boarding station, it should be above ground level. This will mark the end of the sections you designed. Write the function for the fourth section to continue the third function.

For the third in fourth function label:

- the ordered pairs that represent the beginning and end of each of sections.
- all of the peaks (relative maxima) and dips (relative minima).
- the beginning and end of the underground tunnel(s) (
*x*-intercepts/roots) in your design. - The piecewise function that represents the third and fourth sections of your design.

**A roller coaster graph using a 4 piece wise functions**

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