# How do you write the exponential functions goes through the points (1, 6) and (2, 12)?

Apr 7, 2017

$y = 3 \cdot {2}^{x}$

#### Explanation:

All exponential functions can be written in the form $y = a {b}^{x}$. We just need to solve for $a$ and $b$.

Since the function goes through the points $\left(1 , 6\right)$ and $\left(2 , 12\right)$, $x = 1 \implies y = 6$ and $x = 2 \implies y = 12 \setminus \setminus \setminus \setminus \left(1\right)$.

Since $x = 1 \implies y = 6$, we just need to input these values into the equation $y = a {b}^{x}$ to get $6 = a b$.

Since $x = 2 \implies y = 12$, we just need to input these values into the equation $y = a {b}^{x}$ to get $12 = a {b}^{2} \setminus \setminus \setminus \setminus \left(2\right)$.

Since neither $a$ nor $b$ can be zero, we can safely divide equation $\left(2\right)$ by equation $\left(1\right)$ to get $\frac{12}{6} = \frac{a {b}^{2}}{a b}$, or $2 = b$. Since $6 = a b$, it must be the case that $a = 3$.

Our final exponential function is $y = 3 \cdot {2}^{x}$.

We can rewrite this function in several other forms, such as $y = 3 \cdot {2}^{x} = {2}^{\ln} \left(3\right) \cdot {2}^{x} = {2}^{x + \ln \left(3\right)}$ or $y = 3 \cdot {2}^{x} = 3 \cdot {e}^{x \ln \left(2\right)} = 3 \cdot {10}^{x {\log}_{10} \left(2\right)}$.