Admittedly I'm not very creative, but I'm sure you can come up with an interesting name for the roller coaster. Also feel free to try other values of 'a' and other r values (introduce other r values if needed) so you can change the shape of the roller coaster.
I'm using Desmos to graph what is shown in the image attachment below.
The input parameters I used are
[tex]a = -0.1\\r_1 = 0\\r_2 = 10\\r_3 = 15\\r_4 = 20[/tex]
The value of 'a' determines how vertically stretched or compressed the curve will be. Since 0 < |a| < 1 is the case, it's compressed down a bit.
Each r value is a different root where the curve crosses the x axis.
For example, [tex]r_2 = 10[/tex] is where the roller coaster goes into a tunnel.
Notice how this graph has 3 min or max points, aka relative min or relative max. If you want to have 3 such points then you need 3+1 = 4 roots. In general, if you want n min or max points then you need n+1 roots.
There are 4 roots so this is a quartic (not to be confused with a quadratic). Because a < 0, both left and right sides go down forever if we didn't have the domain restriction. But the {0 < x < 20} narrows the domain.
The notation [tex]h(x)\{0 < x < 20\}[/tex] will tell Desmos to graph h(x) only on that specified interval.
