In this blog post, the Desmos activity required that I interact with a racing car app. This racing car app deals with roots and finding which one would pass through the orange cones and to the finish line without going off track. I found it very difficult to find a formula that could accomplish this task. This car racing app had 4 different formulas and locations of the orange cones. The first formula was easy to manipulate in order to this helped you steer the car around the orange cones. I honestly found it really hard to make a formula that goes through the cones and stays on track.

As you can see for the last track with three orange cones I was unable to find a root that stays on track or meets to the finish line. I was able to make it pass through all the orange cones. I am just not sure how to make it stay on track and make it to the finish line.

Can anyone figure out what I could have done differently to make the formula meet the requirements?

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First off, I love that race car gif. I’m old. I used to play that video game as a kid. You’re bringing me back!

It looks like you’ve figured out how to get the roots correct in the formula. The problem is that your graph is too tall so the race car goes off the track. Right now, you have the formula f(x) = Ax(x – 3)(x – 5.5)(x – 8)(x – 11) and in your example, A = 1/20. To make the graph shorter, you need to make A a smaller number. For example, try A = 1/200 so that f(x) = 1/200 x(x – 3)(x – 5.5)(x – 8)(x – 11).

If you want to be really precise, you could pick a point on the track that you want the race car to go through. For example, let’s say you want it to pass through the point (4, 1) so that it just barely goes over the first cone. Then, if you plug in x = 4 into the formula you would get

f(4) = A*4*(4 – 5)(4 – 5.5)(4 – 8)(4 – 11) = A*4*(-1)(-1.5)(-4)(-7) = 168A. But you also know that f(4) = 1 since the graph goes through the point (4, 1). Thus, 168A = 1 and so A = 1/168. See if you can use this method to keep the race car on the track.

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