Questions: Given f(x)=x^3-8, find the equation of the secant line passing through (-3, f(-3)) and (3, f(3)). Write your answer in the form y=mx+b

Solution Steps

To find the equation of the secant line passing through the points \((-3, f(-3))\) and \((3, f(3))\) for the function \(f(x) = x^3 - 8\), follow these steps:

1. Calculate \(f(-3)\) and \(f(3)\).
2. Determine the slope \(m\) of the secant line using the formula \(m = \frac{f(3) - f(-3)}{3 - (-3)}\).
3. Use the point-slope form of the line equation to find the y-intercept \(b\).

Given the function \( f(x) = x^3 - 8 \): \[ f(-3) = (-3)^3 - 8 = -27 - 8 = -35 \] \[ f(3) = 3^3 - 8 = 27 - 8 = 19 \]

The slope \( m \) of the secant line passing through the points \((-3, f(-3))\) and \((3, f(3))\) is calculated as: \[ m = \frac{f(3) - f(-3)}{3 - (-3)} = \frac{19 - (-35)}{3 - (-3)} = \frac{19 + 35}{6} = \frac{54}{6} = 9.0 \]

Using the point-slope form of the line equation \( y = mx + b \) and the point \((3, f(3))\): \[ f(3) = 19 \] \[ 19 = 9.0 \cdot 3 + b \] \[ 19 = 27 + b \] \[ b = 19 - 27 = -8.0 \]

Final Answer