Find the exponential function that passes through the points (2,80) and (5,5120)

  y = 5·4^x

Step-by-step explanation:

If you have two points, (x1, y1) and (x2, y2), whose relationship can be described by the exponential function ...

  y = a·b^x

you can find the values of 'a' and 'b' as follows.

Substitute the given points:

  y1 = a·b^(x1)

  y2 = a·b^(x1)

Divide the second equation by the first:

  y2/y1 = ((ab^(x2))/(ab^(x1)) = b^(x2 -x1)

Take the inverse power (root):

  (y2/y1)^(1/(x2 -x1) = b

Use this value of 'b' to find 'a'. Here, we have solved the first equation for 'a'.

  a = y1/(b^(x1))

In summary:

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For the problem at hand, (x1, y1) = (2, 80) and (x2, y2) = (5, 5120).

  b = (5120/80)^(1/(5-2)) = ∛64 = 4

  a = 80·4^(-2) = 80/16 = 5

The exponential function is ...

  y = 5·4^x

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step-by-step explanation: