Q:

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

Accepted Solution

A:
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 $$y= \frac{1}{2} (4^x)$$.