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I'm trying to create a sliding scale system where if variable $x$ is low, then variable $y$ will be on the lower end of a $1$ to $10$ sliding scale. If variable $x$ is high (above perhaps a threshold?), then variable $y$ will be on the higher end of the sliding scale.

I can't for the life of me figure out the logic behind that. I've tried to do something like:

$$y = \frac{10}{x}$$

But in that case, when $x$ is $100$, $y$ will be low. And when $x$ is $1$, $y$ will be high. I need the exact opposite to happen. I imagine this is a common formula but I cant figure it out. Any ideas?

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@Amzoti that question had a maximum for the input. – Thomas Andrews Dec 12 '12 at 14:07
@ TA: thanks, too early in the morning! – Amzoti Dec 12 '12 at 14:08
up vote 2 down vote accepted

If I'm not mistaken, any function of the form $f(x) = \frac{ax+b}{cx+d}$ should work if $\frac{a}{c} = 10$, $\frac{b}{d}=1$, and $\frac{d}{c} \gt 0$. This will get you a hyperbola with $f(0) = 1$, $\lim\limits_{x \to +\infty} f(x) = 10$, and the vertical asymptote in the negative $x$'s, so it approaches $10$ from below.

For example, $f(x) = \frac{10x+5}{x+5}$ would work fine.

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This solution also seems to work well for these purposes. I'm glad you added the example because I haven't taken a math class in over 10 years (I know, blasphemy) and have no idea what a vertical asymptote is. Thanks for your solution! – Jeremy Harris Dec 12 '12 at 15:43

Would something like $y=10-\frac9{1+x}$ suit you needs?

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Yeah, that seems to work great! – Jeremy Harris Dec 12 '12 at 15:41

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