Answer: [tex]y=(-8)2^x[/tex]
Step-by-step explanation:
The exponential function will have this form:
[tex]y=ab^x[/tex]
We know that the function passes through the points [tex](2,-32)[/tex] and [tex](3,-64)[/tex]. Then, we can substitute the coordinates of the point [tex](2,-32)[/tex] into [tex]y=ab^x[/tex] and solve for "a":
[tex]-32=ab^2\\\\a=\frac{-32}{b^2}[/tex]
Then, we know that:
[tex]y=(\frac{-32}{b^2})b^x[/tex]
Now, we neeed to substitute the coordinates of the second point [tex](3,-64)[/tex] into [tex]y=(\frac{-32}{b^2})b^x[/tex] and solve for "b":
[tex]-64=(\frac{-32}{b^2})b^3\\\\-64=-32b\\\\\frac{-64}{-32}=b\\\\b=2[/tex]
Substituting the value of "b" into [tex]a=\frac{-32}{b^2}[/tex] we can find "a":
[tex]a\frac{-32}{2^2}\\\\a=-8[/tex]
Therefore, we get that the exponential function that describes the graph, is:
[tex]y=(-8)2^x[/tex]
