Question

question
write an exponential function in the form $y = ab^x$ that goes through the points $(0, 13)$ and $(3, 6656)$.
answer attempt 1 out of 3

Explanation:

Step1: Find the value of \(a\)

We know that the exponential function is \(y = ab^x\) and it passes through the point \((0, 13)\). Substitute \(x = 0\) and \(y = 13\) into the function:
\(13 = ab^{0}\)
Since any non - zero number to the power of \(0\) is \(1\) (\(b^{0}=1\) for \(b
eq0\)), we have \(13=a\times1\), so \(a = 13\).

Step2: Find the value of \(b\)

Now that we know \(a = 13\), the function becomes \(y=13b^{x}\). The function also passes through the point \((3, 6656)\). Substitute \(x = 3\), \(y = 6656\) and \(a = 13\) into the function:
\(6656=13\times b^{3}\)
First, divide both sides of the equation by \(13\):
\(\frac{6656}{13}=b^{3}\)
\(512 = b^{3}\)
We know that \(8^{3}=512\) (because \(8\times8\times8 = 512\)), so \(b = 8\).

Step3: Write the exponential function

Now that we have \(a = 13\) and \(b = 8\), the exponential function in the form \(y = ab^{x}\) is \(y = 13\times8^{x}\).

Answer:

\(y = 13\times8^{x}\)