Find the Exponential Function (2,12)

$(2, 12)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $12$ of the point, and set $x$ to the $x$ value $2$ of the point.

$12 = a^{2}$

Step 2

Solve the equation for $a$ .

Rewrite the equation as $a^{2} = 12$ .

$a^{2} = 12$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{12}$

Simplify $\pm \sqrt{12}$ .

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$a = \pm \sqrt{4 (3)}$

Rewrite $4$ as $2^{2}$ .

$a = \pm \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$a = \pm 2 \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$a = 2 \sqrt{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$a = - 2 \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$a = 2 \sqrt{3}, - 2 \sqrt{3}$

Step 3

Substitute each value for $a$ back into the function $f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(2 \sqrt{3})}^{x}, f (x) = {(- 2 \sqrt{3})}^{x}$

Find the Exponential Function (2,12)

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