Ensure exponential function passes through two specific points

Question

I made this function and I'd like to know how can I make sure the function passes through (0,0) and (1,1) and how can I mirror it so I have the same function but inverted that goes downward instead of upward and passes through (0,0) and (1,-1) ?

All I know about the exponential function is that its equation is f(x) = ab^x+h+k where h controls the horizontal shifting and k controls the vertical shifting. I might be wrong though.

Note: I have a background in computer programming and I don't know the mathematical terms very well. I apologize for my lack of knowledge.

Vishu · Accepted Answer · 2020-04-19T20:00:55.733

1

You want $f(1) = 1$, which is equivalent to saying $ab^{h+1} +k=1$ and $f(0)=0 \implies ab^h + k=0$. Any combination of $a,b,h,k$ that satisfies these equations will be good to go (there are infinitely many).

You can invert this function w.r.t to the $x$-axis just by adding a negative sign. So $f(x)$ will be $-(ab^{x+h} + k)$.

Eliminating $k$ from the two equations, we get $$ab^h(b-1)=1$$

As an example, $a=1, b=2, h=0, k=-1$ satisfies these conditions.

To generate more solutions, assign $h$ and $k$ arbitrary values and then solve for $a$ and $b$ which should be easy to do.

edited Apr 19 '20 at 20:00

answered Apr 19 '20 at 18:48

Vishu

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But how can I find a combination of a, b, h, k that satisfies these equations? An example would be greatly appreciated. – Elie G. Apr 19 '20 at 19:01
@ElieG. Edited. – Vishu Apr 19 '20 at 20:01

Ensure exponential function passes through two specific points

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