Question

Write an exponential function in the form y = a that goes through points ( 0 , 13 ) and ( 2 , 637 )

Answer 1

`$ y = (13)7^x $`

Explanation:

When two points are given say,
`$ (x_1, y_1) $` and
`$ (x_2, y_2) $` the exponential function passing through these points is given by

`$ y = ab^x $`

This is obtained by solving for
`$ a $` and
`$ b $` in

`$ y_1 = ab^(x_1) \hspace{15mm} (1) $`

`$ y_2  = ab^(x_2) \hspace{15mm} (2) $`

when we solve (1) and (2) for
`$ a $` and
`$ b $` we substitute it in the main equation to get the desired exponential function.

Here:
`$ (x_1, y_1) = (0, 13) $` and
`$ (x_2, y_2) = (2, 637) $`.

Substituting in (1) and (2), we get:

`$ 13 = ab^0 $`

`$ 637 = ab^2 $`

From the first equation we get:
`$ a = 13 $` [since
`$ b^0 = 1 $`].

Now substitute a = 13 in the second equation to get the value of b.

`$ \implies 637 = (13)b^2$`

`$ \implies b^2 = 49 $`
`$ \implies b = 7 $`

Substituting the values in the main equation we get:

`$  y = (13)7^x $`