To write an exponential function in the form $y = ab^x$ that goes through the points $(0, 14)$ and $(3, 1750)$, follow these steps:

Step 1: Use the first point $(0, 14)$

Substitute $x = 0$ and $y = 14$ into the equation $y = ab^x$:

$14 = ab^0$

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

$14 = a \cdot 1 \quad \Rightarrow \quad a = 14$

So, the equation becomes:

$y = 14b^x$

Step 2: Use the second point $(3, 1750)$

Now, substitute $x = 3$ and $y = 1750$ into the equation $y = 14b^x$:

$1750 = 14b^3$

Solve for $b$:

$b^3 = \frac{1750}{14} = 125 \quad \Rightarrow \quad b = \sqrt[3]{125} = 5$

Step 3: Final equation

Now that we have $a = 14$ and $b = 5$, the exponential function is:

$y = 14 \cdot 5^x$

Final Answer:

The exponential function is:

$y = 14 \cdot 5^x$

Would you like further details on how this was solved, or do you have any other questions? Here are a few more related questions:

1. How do you graph an exponential function like $y = 14 \cdot 5^x$?
2. How can you solve for $b$ if you're given different points?
3. What are the properties of an exponential function like this one?
4. How do you find the rate of growth for an exponential function?
5. How can you apply this type of function in real-world scenarios?

Tip: Always check if a point is on the graph by plugging it into the equation!