Answer:
[tex]y = 0.47(0.352)^x[/tex]
Step-by-step explanation:
Fun with exponential functions! :D
So, we know that our goal is to write an equation that looks like
[tex]y = ab^x[/tex]
To do this, we need to figure out what "a" and "b" are equal to. How can we do that? By plugging in the coordinates and writing some equations.
The first coordinate is (-3, 10.8). This says that x = -3 and y = 10.8. Let's plug that into our equation:
[tex]y = ab^x\\10.8 = ab^{-3}[/tex]
Hmm... okay, that's interesting, but not sure what to do with that yet. Let's plug in the second coordinate and see what happens. (-2, 3.6) says that x = -2 and y = 3.6.
[tex]y = ab^x\\3.8 = ab^{-2}[/tex]
So, we have two equations now:
[tex]Eq 1) \quad 10.8 = ab^{-3}\\Eq 2) \quad 3.8 = ab^{-2}[/tex]
Let's solve the first equation for "a" by getting "a" by itself. We divide by [tex]b^{-3}[/tex]
[tex]10.8=ab^{-3}\\\frac{10.8}{b^{-3}}=\frac{ab^{-3}}{b^{-3}}\\10.8 b^3 = a[/tex]
Then, plug this into the second equation:
[tex]3.8 = ab^{-2}\\3.8 = (10.8b^3)b^{-2}\\3.8 = 10.8b\\\frac{3.8}{10.8} = b\\0.352 \approx b[/tex]
Yay! We've found what "b" equals. Now, what about "a"? We can just plug the value for "b" back into the equation we made for "a":
[tex]10.8b^3 = a\\10.8(\frac{3.8}{10.8})^3 = a\\0.470 \approx a[/tex]
So, our final equation is
[tex]y = 0.47(0.352)^x[/tex]
