Question

write an exponential function in the form $y = ab^x$ that goes through the points $(0, 2)$ and $(3, 128)$.

Explanation:

Step1: Find the value of \(a\) using the point \((0, 2)\)

Substitute \(x = 0\) and \(y = 2\) into the equation \(y = ab^{x}\). We know that any non - zero number to the power of \(0\) is \(1\), so \(b^{0}=1\). Then the equation becomes \(2=a\times b^{0}\), which simplifies to \(2 = a\times1\), so \(a = 2\).

Step2: Find the value of \(b\) using the point \((3, 128)\) and \(a = 2\)

Now that we know \(a = 2\), substitute \(x = 3\), \(y = 128\) and \(a = 2\) into the equation \(y=ab^{x}\). We get \(128 = 2\times b^{3}\). First, divide both sides of the equation by \(2\): \(\frac{128}{2}=\frac{2\times b^{3}}{2}\), which gives \(64=b^{3}\). Then, take the cube root of both sides. Since \(4^{3}=64\), we have \(b = 4\).

Step3: Write the exponential function

Now that we have found \(a = 2\) and \(b = 4\), substitute these values back into the general form of the exponential function \(y = ab^{x}\). So the function is \(y=2\times4^{x}\).

Answer:

\(y = 2\times4^{x}\)