Given \( f(x)=x^{2}+2 x \), find the equation of the secant line passing through \( (-7, f(-7)) \) and \( (1, f(1) \) ). Write your answer in the form \( y=m x+b \).

To find the equation of the secant line through the points \((-7, f(-7))\) and \((1, f(1))\), we first need to evaluate \(f(-7)\) and \(f(1)\). 1. Calculate \(f(-7)\): \[ f(-7) = (-7)^{2} + 2(-7) = 49 - 14 = 35 \] So the point \((-7, f(-7))\) is \((-7, 35)\). 2. Calculate \(f(1)\): \[ f(1) = (1)^{2} + 2(1) = 1 + 2 = 3 \] So the point \((1, f(1))\) is \((1, 3)\). Now we have the two points: \((-7, 35)\) and \((1, 3)\). Next, we need to find the slope \(m\) of the secant line, which is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where \((x_1, y_1) = (-7, 35)\) and \((x_2, y_2) = (1, 3)\): \[ m = \frac{3 - 35}{1 - (-7)} = \frac{3 - 35}{1 + 7} = \frac{-32}{8} = -4 \] Now we have the slope \(m = -4\). We can use the point-slope form of the equation of a line \(y - y_1 = m(x - x_1)\). Using the point \((1, 3)\): \[ y - 3 = -4(x - 1) \] Expanding this: \[ y - 3 = -4x + 4 \] \[ y = -4x + 4 + 3 \] \[ y = -4x + 7 \] Thus, the equation of the secant line in the form \(y = mx + b\) is: \[ \boxed{y = -4x + 7} \]