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The problem asks to find the slope of the secant line for the function \( f(x) = 1.2x^3 - 32x^2 + 261x + 556 \), between \( x_1 = 0 \) and \( x_2 = 4 \).

The formula for the slope of a secant line between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is:

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

### Step 1: Evaluate \( f(0) \)
\[
f(0) = 1.2(0)^3 - 32(0)^2 + 261(0) + 556 = 556
\]

### Step 2: Evaluate \( f(4) \)
\[
f(4) = 1.2(4)^3 - 32(4)^2 + 261(4) + 556
\]
Let me calculate this value.After evaluating \( f(4) \), we get \( f(4) = 1164.8 \).

### Step 3: Calculate the slope of the secant line
Using the slope formula:

\[
m = \frac{f(4) - f(0)}{4 - 0} = \frac{1164.8 - 556}{4} = \frac{608.8}{4} = 152.2
\]

Rounding to the nearest whole number, the slope is \( 152 \).

Would you like more details, or do you have any questions?

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Here are five related questions to expand your understanding:
1. How do you calculate the slope of a tangent line at a specific point on a curve?
2. What is the geometric interpretation of the slope of a secant line?
3. Can the secant line slope be negative? If yes, under what conditions?
4. How would the slope change if we considered \( x_2 = 6 \) instead of \( x_2 = 4 \)?
5. How does the cubic term in \( f(x) = 1.2x^3 - 32x^2 + 261x + 556 \) affect the shape of the curve?

**Tip:** The slope of a secant line represents the average rate of change of the function between two points.

The function f(x) = 1.2x^3 - 32x^2 + 261x + 556 models the number of discharges from the military, f(x), of active-duty gay service members under the 'don't ask, don't tell' policy x years after 1994. Find the slope of the secant line from x₁ = 0 to x₂ = 4.

Calculate the slope of the secant line for the cubic function f(x) = 1.2x^3 - 32x^2 + 261x + 556, which models the number of military discharges after 1994. Detailed solution and key algebraic concepts included.

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