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`(1,3) , (2,12)` Write an exponential function `y=ab^x` whose graph passes through the given points.

The given two points of the exponential function are (1,3) and (2,12).


To determine the exponential function


`y=ab^x`


plug-in the given x and y values.


For the first point (1,3), plug-in x = 1 and y=3.


`3=ab^1`


`3=ab`        (Let this be EQ1.)


For the second point (2,12), plug-in x=2 and y=12.


`12=ab^2`     (Let this be EQ2.)


To solve the values of a and b, apply the substitution method of system of equations. To do so, isolate the a in EQ1.


`3=ab`


`3/b= a`


Plug-in this to EQ2.


`12=ab^2`


`12=(3/b)b^2`


And, solve for b.


`12=3b`


`12/3=b`


`4=b`


Plug-in this value of b to EQ1.


`3=ab`


`3=a(4)`


And, solve for a.


`3/4=a`


Then, plug-in the values of a and b to the exponential function


`y=ab^x`


So, this becomes:


`y = 3/4*4^x`


Therefore, the exponential function that passes the given two points is `y=3/4*4^x` .

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