We are tasked with writing an exponential function of the form:

$y = ab^x$

that passes through the points $(0, 11)$ and $(3, 3773)$.

Step 1: Using the first point $(0, 11)$

Substitute $x = 0$ and $y = 11$ into the equation $y = ab^x$:

$11 = ab^0$

Since $b^0 = 1$, this simplifies to:

$11 = a$

So, $a = 11$.

Step 2: Using the second point $(3, 3773)$

Now substitute $x = 3$ and $y = 3773$ into the equation $y = ab^x$, using $a = 11$:

$3773 = 11b^3$

Now, solve for $b^3$:

$b^3 = \frac{3773}{11} = 343$

Next, solve for $b$:

$b = \sqrt[3]{343} = 7$

Step 3: Write the final equation

Now that we have $a = 11$ and $b = 7$, the exponential function is:

$y = 11 \cdot 7^x$

Final Answer:

$y = 11 \cdot 7^x$

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Would you like further details or explanations on any part of the process? Here are some related questions to expand your understanding:

1. What is the general form of an exponential function and how is it derived?
2. How can we graph exponential functions and determine their behavior?
3. What happens to the graph of an exponential function if $a$ or $b$ is negative?
4. How can you solve for the unknown parameters of an exponential function with more than two points?
5. How do logarithms help in solving equations involving exponents?

Tip: When working with exponential functions, always make sure to check your work by substituting the given points back into the equation to ensure it holds true!