Question

Write an exponential function whose graph passes through the given points (2,8) (5,512)

Answer 1

We are going to use the basic form of an exponential function: $y=ab^x$ to solve this problem.
Since we have tow points $(x_{1},y_{1})$ and $(x_{2},y_{2})$, we are going to have a system of equations:
$y_{1}=ab^{x_{1}$ equation (1)
$y_{2}=ab^{x_{2}}$ equation (2) 

We know for our problem that $x_{1}=2$, $y_{1}=8$, $x_{2}=5$, and $y_{2}=512$, so lets replace those values in our equations: 
$8=ab^2$ equation (1)
$512=ab^5$ equation (2)

Now, we just need to find $a$ and $b$:

Dividing equation (2) by equation (1):
$\frac{512=ab^5}{8=ab^2}$
$64=b^3$
$b= \sqrt[3]{64}$
$b=4$ equation (3)

Replacing equation (3) in equation (1):
$8=a(4)^2$
$8=16a$
$a= \frac{8}{16}$
$a= \frac{1}{2}$

Finally, we can put our function together:
$y=ab^x$
$y= \frac{1}{2} (4^x)$

We can conclude that the exponential function whose graph passes through the points (2,8) and (5,512) is $y= \frac{1}{2} (4^x)$.