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To find the slope of the secant line between the points \( x_1 = 0 \) and \( x_2 = 4 \), we need to apply the formula for the slope of a secant line:

\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

From the graph, the function is given by:

\[
f(x) = 1.3x^3 - 34x^2 + 270x + 574
\]

We will evaluate the function at \( x_1 = 0 \) and \( x_2 = 4 \):

- \( f(0) \)
- \( f(4) \)

Let me calculate the values and then find the slope.The values for the function at \(x_1 = 0\) and \(x_2 = 4\) are:

- \( f(0) = 574 \)
- \( f(4) = 1193.2 \)

Now, using the formula for the slope of the secant line:

\[
m = \frac{f(4) - f(0)}{4 - 0} = \frac{1193.2 - 574}{4} = 154.8
\]

Rounding to the nearest whole number, the slope of the secant line is **155**.

Would you like more details or have any questions? Here are some related questions for deeper exploration:

1. What does the slope of the secant line represent in the context of this problem?
2. How would you find the derivative of the function to get the slope of the tangent line at a specific point?
3. What is the meaning of the function’s cubic term in real-life terms for the discharge rate?
4. How does the shape of the graph relate to the changes in the discharge numbers over time?
5. What would happen to the secant line's slope if we calculated it over a different interval?

**Tip:** The secant line approximates the average rate of change of the function over the interval.

Find the slope of the secant line from x1 = 0 to x2 = 4 for the function f(x) = 1.3x^3 - 34x^2 + 270x + 574, which models the number of military discharges after the 'don't ask, don't tell' policy in years after 1994.

Learn how to calculate the slope of a secant line for the cubic function f(x) = 1.3x^3 - 34x^2 + 270x + 574, which models the number of military discharges after the 'don't ask, don't tell' policy in years after 1994. Detailed step-by-step solution suitable for students at the high school or college level.

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