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I am using a Casio graphing calculator's Picture Plot feature.

I am trying to find the equation which models a rollercoaster.

I have plotted the equation as seen below:

my plot

Through quartic regression, I'm able to get the equation:

$$0.31~x^4 -0.38~x^3 - 2.5~x^2 + 0.52$$

Though this is an equation in which each interval (each mark on the axes) represents 1 unit. In reality the roller coaster section seen has a height of 107m and a length of 20m (length is hypothetical).

I'm unable to realize how to change the equation to account for this.

Matti P.
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Math Comorbidity
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    The part of the graph that you're outlining, reaches its highest point when $x=0$, in other words the heights is $0.52$. So if you want to make the height equal to $107$, you simply multiply the equation by $\frac{107}{0.52}$. As for the length, that will be much trickier because the evaluation of the graph length requires a tricky integral ... – Matti P. Oct 12 '20 at 10:16
  • Just to clarify, what do you mean by "length" in this context? Is it the width of the part that you have drawn? Or is it really the length of the section, length of the graph? – Matti P. Oct 12 '20 at 10:17
  • Here's a graph of the function: https://www.desmos.com/calculator/emtdgzo1pf As you can see, it has several peaks and valleys. One would have to clarify which one we want to use. – Matti P. Oct 12 '20 at 10:18
  • @MattiP. what I meant to say was that when I perform regression for the plotted points, the points have been plotted where the axes have steps of 1 each therefore when the equation is produced it is not representative of the actual rollercoaster. I want to account for the rollercoasters real height and width which I'm unable to do through the calculator so was wondering if there is a way by hand (for example multiplying each coordinate by a value etc) – Math Comorbidity Oct 12 '20 at 10:19
  • Perhaps instead of $$0.31~x^4 -0.38~x^3 - 2.5~x^2 + 0.52$$ you mean $$-0.31x^4 +0.38~x^3 + 2.5~x^2 - 0.52$$ and then you're interested in the interval from $x\approx 0.5$ to $x \approx 3.5$? – Matti P. Oct 12 '20 at 10:20
  • @MattiP. I've realized that I might've made a mistake in the regression. Though what I want to know is that if I have points plotted in a range of for [4,5] where 5 represents the value of 107 meters, and a domain of [0,5] where 5 represents 20 meters, how do I change the equation – Math Comorbidity Oct 12 '20 at 10:26
  • So you want to scale a range of numbers from one scale to another? Here's how: https://math.stackexchange.com/questions/3355027/a-function-or-a-factor-to-scale-a-list-of-real-numbers-from-one-range-to-another/3355071#3355071 – Matti P. Oct 12 '20 at 10:32
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    It's not entirely clear for me, how these ranges $[4,5]$ and $[0,5]$ are manifested in the equations that you have given ... So please check your regression and come back with the updated results. Then I can look at it and give you some hints, ok? – Matti P. Oct 12 '20 at 10:35
  • @MattiP. Thanks a lot for that! I'll look at that and get back. As I'm relatively new, I wanted to know if I solve my question through your comments, should I delete my question or leave it as it is? – Math Comorbidity Oct 12 '20 at 11:13

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