Find the formula for an exponential function that passes through the two points given. \( (0,6) \) and \( (4,3750) \)

To find the formula for an exponential function that passes through the points (0, 6) and (4, 3750), we can use the general form of an exponential function: \( f(x) = ab^x \). From the first point (0, 6), we can see that when \( x = 0 \), \( f(0) = ab^0 = a = 6 \). Thus, we have \( a = 6 \). Now we can use the second point (4, 3750) to find \( b \). Substituting the values into the function gives us \( 3750 = 6b^4 \). Rearranging this equation, we get \( b^4 = \frac{3750}{6} = 625 \). Taking the fourth root of both sides, we get \( b = \sqrt[4]{625} = 5 \). Therefore, the exponential function is \( f(x) = 6(5^x) \). For further understanding, exponential functions are a key concept in mathematics and can model various real-world scenarios. For instance, they are commonly used to describe population growth, radioactive decay, or even the spread of a virus! If you're looking to improve your skills in working with exponential functions, it's essential to practice more examples. Start with simple functions and gradually increase their complexity. Also, check your calculations at every step because misplacing a decimal or a power can lead to wildly different results!