To find the equation of the secant line passing through the points $(-8, f(-8))$ and $(2, f(2))$ for the function $f(x) = x^2 + 4x$, let's proceed with the following steps:

Step 1: Find $f(-8)$ and $f(2)$

1. Calculate $f(-8)$: $f(-8) = (-8)^2 + 4(-8) = 64 - 32 = 32$ So, the point $(-8, f(-8))$ is $(-8, 32)$.

2. Calculate $f(2)$: $f(2) = (2)^2 + 4(2) = 4 + 8 = 12$ So, the point $(2, f(2))$ is $(2, 12)$.

Step 2: Calculate the Slope of the Secant Line

The slope $m$ of the line passing through $(-8, 32)$ and $(2, 12)$ is given by: $m = \frac{f(2) - f(-8)}{2 - (-8)} = \frac{12 - 32}{2 + 8} = \frac{-20}{10} = -2$

Step 3: Write the Equation of the Secant Line

Now that we know the slope $m = -2$, we can use the point-slope form of the line equation, $y - y_1 = m(x - x_1)$, and choose one of the points, say $(2, 12)$.

Using $m = -2$ and $(x_1, y_1) = (2, 12)$: $y - 12 = -2(x - 2)$ Expanding this: $y - 12 = -2x + 4$ $y = -2x + 16$

Final Answer

The equation of the secant line is: $y = -2x + 16$

Would you like further clarification on any of the steps?

Here are some related questions to consider:

1. How do you find the slope of a line given two points?
2. What is the point-slope form of a line, and how is it used?
3. Why is it called a "secant line" in this context?
4. What is the difference between a secant line and a tangent line?
5. How does the secant line approximate the average rate of change of a function over an interval?

Tip: When finding the equation of a line, carefully choose the correct points to avoid calculation errors, especially with signs.