Find a formula for the exponential function passing through the points (-1,1/2) and (1,8)

Question

asked Jan 8, 2023 200k views

1 Answer

For Comment · Answer 1 · 2023-01-12T11:51:36+0000

The general form of an exponential function is:

We are given the points:

$(-1,(1)/(2))\text{ }and\text{ }(1,8)$

This means:

$\begin{gathered} f(-1)=(1)/(2) \\ . \\ f(1)=8 \end{gathered}$

Thus, we can write:

$\begin{cases}(1)/(2)=ab^(-1){} \\ 8=ab^1\end{cases}$

This is a system of two equations with two variables. We can solve for b in the first equation:

$\begin{gathered} (1)/(2)=ab^(-1) \\ . \\ (1)/(2)=(a)/(b) \\ . \\ b=2a \end{gathered}$

And now, substitute in the second equation:

$\begin{gathered} 8=a(2a)^1 \\ . \\ 8=2a^2 \\ . \\ (8)/(2)=a^2 \\ . \\ a=\pm√(4) \\ . \\ a=\pm2 \end{gathered}$

Now, we can pick any of the two solutions for a, in this case, we'll use the positive one, a = 2. And now we can find b:

$\begin{gathered} b=2\cdot2 \\ . \\ b=4 \end{gathered}$

Now, we can write the equation:

$f(x)=2\cdot4^x$

And we can verify that the points given lie in the graph of f(x), by evaluating:

$\begin{gathered} f(-1)=2\cdot4^(-1)=(2)/(4)=(1)/(2) \\ . \\ f(1)=2\cdot4^1=2\cdot4=8 \end{gathered}$

Thus, the answer:

$f(x)=2\cdot4^x$

is correct.