Question

question help:

find a formula for the exponential function passing through the points (-3,1/2) and (1,8).

y=

Explanation:

Step1: Write the general form of exponential function

The general form of an exponential function is \(y = ab^{x}\).

Step2: Substitute the given points into the general form

Substitute \((x=-3,y = 2)\) into \(y=ab^{x}\), we get \(2 = ab^{-3}=\frac{a}{b^{3}}\). Substitute \((x = 1,y=8)\) into \(y=ab^{x}\), we get \(8=ab^{1}=ab\).

Step3: Solve the system of equations

From \(8 = ab\), we have \(a=\frac{8}{b}\). Substitute \(a=\frac{8}{b}\) into \(\frac{a}{b^{3}}=2\), then \(\frac{\frac{8}{b}}{b^{3}}=2\), which simplifies to \(\frac{8}{b^{4}}=2\). Cross - multiply gives \(2b^{4}=8\), so \(b^{4}=4\), and \(b = \sqrt{2}\) (since \(b>0\) for an exponential function \(y = ab^{x}\)).
Since \(a=\frac{8}{b}\) and \(b=\sqrt{2}\), then \(a=\frac{8}{\sqrt{2}} = 4\sqrt{2}\).

Answer:

\(y = 4\sqrt{2}\times(\sqrt{2})^{x}\)