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To solve this, we're given an exponential function \( y = ab^x \) and two points from the graph: (10, 320) and (30, 20). We need to determine the values of \(a\) and \(b\), then round \(a\) to the nearest integer and \(b\) to three decimal places.

### Step-by-step solution:

1. **Start with the general exponential form:**
   \[
   y = ab^x
   \]
   Given two points:
   \[
   (x_1, y_1) = (10, 320), \quad (x_2, y_2) = (30, 20)
   \]
   Substituting these into the equation will give us a system of two equations.

2. **Substitute the first point (10, 320):**
   \[
   320 = ab^{10}
   \]

3. **Substitute the second point (30, 20):**
   \[
   20 = ab^{30}
   \]

4. **Solve the system of equations:**
   Divide the second equation by the first to eliminate \(a\):
   \[
   \frac{20}{320} = \frac{ab^{30}}{ab^{10}}
   \]
   Simplify:
   \[
   \frac{1}{16} = b^{20}
   \]
   Take the 20th root of both sides to solve for \(b\):
   \[
   b = \left( \frac{1}{16} \right)^{\frac{1}{20}} = 16^{-\frac{1}{20}}
   \]

   Now, we calculate \(b\).

5. **Find \(a\):**
   Once \(b\) is determined, substitute it back into one of the original equations (e.g., \(320 = ab^{10}\)) to solve for \(a\).

Let me now compute these values.The formula for the exponential function is:

\[
y = 1280 \cdot 0.871^x
\]

Where:
- \( a = 1280 \) (rounded to the nearest integer)
- \( b = 0.871 \) (rounded to three decimal places)

Do you have any questions or would you like further details?

### Here are 5 related questions:
1. How do exponential functions behave for negative values of \(x\)?
2. Can exponential functions ever result in zero? Why or why not?
3. How can you determine the rate of decay from an exponential function?
4. What is the significance of the base \(b\) in the context of growth and decay?
5. How do logarithms relate to solving exponential equations?

### Tip:
When working with exponential functions, the base \(b\) (if \(0 < b < 1\)) represents decay, and if \(b > 1\), it represents growth.

Find a formula for the exponential function, given points (10, 320) and (30, 20). Round a to the nearest integer and b to three decimal places.

Learn how to find an exponential function formula given points (10, 320) and (30, 20) with step-by-step calculations. This problem covers exponential function formulas and system of equations, suitable for students in Grades 9-12.

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