equation from exponential graph

Question

I am struggling to get an equation from the graph above

I know

${f(x) = ba^x + c}$

When y = 0, x = -2

${b = -2}$

${24 = -2a^3}$

${2a^3 = -24}$

${a = {\sqrt[3]{{-24}\over 2}}}$

Answer 1

There are different models that you can use to express exponential functions. A common one is:

$$y = ba^x$$

Though you can also use the model $y = ba^x + c$ as you have proposed, this model includes three unknown parameters. Thus, to find an equation fitting this model, you would need to be given three points. However, you are only given two points, so this model will not work here.

So, assuming we want to use the common model I mentioned above, we note that on the given curve when $x=0$, $y=-2$. Plugging these values into the formula we find that $b=-2$.

Using this result, you have already shown that you can then figure out

$$a=\sqrt[3]{\frac{-24}{2}}$$

Note, however, that this is not the simplest form, so you should simplify it. Now, all that remains is to plug the parameters $b$ and $a$, which you now know, back into the general formula $y=ba^x$.

As you can see, you have done most of the work to get to the correct answer, but I hope I've cleared up some confusion on the process of getting the correct result.