A Math Project
Build a Roller Coaster
with Equations
Every thrilling drop, every soaring hill, every tight turn — all of it is just math. In this project, you'll use real functions to design a 2D roller coaster track, then watch your coaster ride it.
How it works
01
Write Equations
Each piece of your track is a mathematical function — like a parabola for a hill, a sine wave for a wave section, or a linear function for a ramp. You control the domain of each piece.
02
Connect the Pieces
Real roller coasters can't have gaps or jumps. You'll need to make sure your functions connect smoothly — the end of one segment must match the start of the next. The app tells you if something's disconnected.
03
Ride It
Once your track is built, hit Launch and watch a coaster car travel along the path your equations created. The steeper the slope, the faster it goes!
04
Use Division to Go Deeper
Want to create more interesting curves? Use the Synthetic Division Explorer to find roots of polynomials and understand exactly where your track crosses zero — the ground level.
Your Challenge
Design a Roller Coaster that Follows These Rules
Your coaster must use at least 4 different function segments, include at least one hill (a parabola or similar curve that goes up and comes back down), and start and end at ground level (y = 0). Every segment must connect to the next with no gaps or jumps.
📐
Ramps Use linear functions like y = 2x to create slopes. Steeper coefficient = steeper ramp.
🏔️
Hills & Valleys Parabolas like y = -(x-4)²+6 make perfect arched hills. Flip the sign for a valley.
🌊
Wave Sections y = 2·sin(x) creates smooth oscillations — great for a wave section of the coaster.
⛺
Flat Sections Simple y = 0 or y = 3 give you level track for loading or transitions.
💡 Want Smarter Curves?
Use Synthetic Division to Find Your Track's Roots
When your track segment is a polynomial like x³ − 6x² + 11x − 6, synthetic division helps you quickly find where it equals zero — those are the exact x-values where your track touches the ground.

That means you can design a hill that lands precisely at ground level, connecting cleanly to the next segment. No guessing, no ugly gaps.
▶ x³ − 6x² + 11x − 6 has roots at x = 1, 2, 3
▶ x³ + 8 has a root at x = −2
▶ 2x³ + 3x² − 5x + 2 remainder at x=1 tells you p(1)
EXAMPLE
2 1 −6 11 −6
2 −8 6
1 −4 3 0
x=2 is a root → remainder 0
quotient: x² − 4x + 3
🎢
Roller Coaster Builder
Design your track with equations — then ride it
Module 1
Viewport
x min
x max
y min
y max
Track Segments
Sample Tracks
Click to:
x: 0.000 y: 0.000
READY
CARS
SPEED
1.0x
Lead Car X
--
Lead Car Y
--
Slope
--
Segments
0
Track Length
0.00
➗
Synthetic Division Explorer
Step-by-step polynomial division — see every move
Module 2
Polynomial Setup
Degree:
Coefficients (leading → constant)
Divide by (x − r), enter r
r =
—
Sample Problems
Division Table
Coefficient
Bring Down / Sum
Product (r × above)
Quotient Coeff
Remainder
Set up a polynomial above and press Start.
Step-by-Step Explanation
Steps will appear here as you work through the division.
Result
Remainder Theorem — Visual
p(r) = remainder
x window −8 to 8
Track domain: −8 to 8
Enter a polynomial and divisor to see the graph.
Roller Coaster Segment Preview
Drag the domain handles above to select a track segment.