Write an exponential function in the form y, equals, a, b, to the power xy=ab 
x
  that goes through the points left parenthesis, 0, comma, 17, right parenthesis(0,17) and left parenthesis, 3, comma, 8704, right parenthesis(3,8704).

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

\[
y = ab^x
\]

that passes through the points \((0, 17)\) and \((3, 8704)\).

### Step 1: Use the point \((0, 17)\)
When \(x = 0\), we substitute into the exponential equation:

\[
17 = ab^0
\]

Since \(b^0 = 1\), we get:

\[
17 = a
\]

Thus, the equation becomes:

\[
y = 17b^x
\]

### Step 2: Use the point \((3, 8704)\)
Now, substitute \(x = 3\) and \(y = 8704\) into the updated equation:

\[
8704 = 17b^3
\]

### Step 3: Solve for \(b\)
To find \(b\), divide both sides by 17:

\[
b^3 = \frac{8704}{17} = 512
\]

Now, solve for \(b\) by taking the cube root of 512:

\[
b = \sqrt[3]{512} = 8
\]

### Step 4: Write the final equation
Now that we know \(a = 17\) and \(b = 8\), the exponential function is:

\[
y = 17 \cdot 8^x
\]

This is the exponential function that passes through the points \((0, 17)\) and \((3, 8704)\).

---

Would you like any further details or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you graph an exponential function like \(y = 17 \cdot 8^x\)?
2. What is the general effect of increasing or decreasing \(b\) in the equation \(y = ab^x\)?
3. How would the equation change if the point \((3, 8704)\) were different?
4. Can exponential functions model real-world situations? What are some examples?
5. How does an exponential function compare to a linear function in terms of growth?

**Tip**: When solving for parameters like \(b\), always remember that exponentiation is reversible through roots or logarithms.

Write an exponential function in the form y = ab^x that goes through the points (0, 17) and (3, 8704).

Learn how to find an exponential function y = ab^x that passes through the points (0, 17) and (3, 8704). This problem covers key concepts in algebra and exponential functions. Suitable for students in Grades 9-12.

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