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) = abx+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.
Best Answer
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.