Question

Find an equation for an exponential function that passes through the pair of points given. (Round all coefficients to 4 decimal places when necessary.) HINT [See Example 2.] Through (1, 5) and (3, 15)

Answer 1

The exponential function is:

`\[ f(x) = 5 * 3.1623^(2x) \]`

Step-by-step explanation:

To find the equation for the exponential function passing through the points (1, 5) and (3, 15), we use the general form
`\(f(x) = a * b^x\), where \(a\) is the initial value and \(b\) is the growth factor. Substituting the given points, we get two equations: \(5 = a * b\) and \(15 = a * b^3\). Solving these simultaneously, we find \(a = 5\) and \(b \approx 3.1623\). Therefore, the exponential function is \(f(x) = 5 * 3.1623^(2x)\),` rounding the coefficient to four decimal places.

This equation represents exponential growth because the base (\(b\)) is greater than 1. The initial value (\(a = 5\)) is the value of the function when \(x = 0\), and the growth factor (\(b\)) determines how quickly the function increases. In this case, the function starts at 5 and grows rapidly with each increase in \(x\). The points (1, 5) and (3, 15) satisfy this equation, confirming its accuracy. The rounding of coefficients to four decimal places ensures a concise and manageable representation of the exponential function.