To solve the problem, we are given the exponential function $f(x) = ab^x$, and it passes through the points $(0, 4000)$ and $(3, 1372)$. We need to find the values of $a$ and $b$.

Step 1: Solve for $a$ using the point $(0, 4000)$

When $x = 0$: $f(0) = ab^0 = a \cdot 1 = a$ From the point $(0, 4000)$, we know: $a = 4000$

Step 2: Substitute $a = 4000$ into the equation

The function becomes: $f(x) = 4000b^x$

Step 3: Use the point $(3, 1372)$ to solve for $b$

When $x = 3$, $f(3) = 1372$: $1372 = 4000b^3$ Divide through by 4000: $b^3 = \frac{1372}{4000}$ $b^3 = 0.343$

Step 4: Solve for $b$

Take the cube root of both sides: $b = \sqrt[3]{0.343} = 0.7$

Final Answers:

$a = 4000, \quad b = 0.7$

Let me know if you want detailed steps or further explanations!

Related Questions:

1. How do we interpret the base $b$ in the context of exponential growth or decay?
2. Can we find the equation of the exponential function after determining $a$ and $b$?
3. What would the function predict for $x = 5$?
4. How would the process differ for other types of functions, such as quadratic functions?
5. What happens if one of the points lies at $x = 1$? How does it simplify solving for $b$?

Tip:

Always double-check your substitution into the function to ensure the given points are satisfied.