Given 4 points on a curve, where the curve is defined as $y = ab^x + c$, how do I solve for a, b and c, if necessary?

Question

I've been working at solving this equation all night and I keep hitting dead-ends. Given the curve can be defined as $y = ab^x + c$ and the points $(-1, 0)$, $(0, 0.01)$, $(0.9, 0.5)$, and ($1, 1)$ exist on it, how do I find constants $a$, $b$ and $c$?

Answer 1

You could solve this with only $3$ coordinates.

Plugging in the coordinates:

$$0=ab^{-1}+c \implies 0=\dfrac ab+c\tag1$$

$$0.01=ab^0+c \implies0.01=a+c\tag2$$

$$0.5=ab^{0.9}+c\tag3$$

$$1=ab^1+c \implies 1=ab+c \tag4$$

There are $4$ equations, $3$ unknowns. Out of those $4$ equations, focus on the $3$ easiest ones to solve (IMO $(1), (2)$, and $(4)$)

Can you solve the $4$ equations, or do you need further hints?