Let \( f(x)=x^{2}+4 x \) (A) Find the slope of the secant line from \( x=3 \) to \( x=9 \). Slope of secant line \( =16 \) (B) Find the equation of the secant line from \( x=3 \) to \( x=9 \). \( y=\square \) (C) Find the slope of the secant line from \( x=3 \) to \( x=6 \). Slope of secant line \( =\square \) (D) Find the equation of the secant line from \( x=3 \) to \( x=6 \). \( y=\square \)

To solve the problems step-by-step, let’s first tackle part (A). (A) To find the slope of the secant line from \( x = 3 \) to \( x = 9 \), we need to calculate the values of the function at these points. 1. Calculate \( f(3) \): \[ f(3) = 3^2 + 4(3) = 9 + 12 = 21 \] 2. Calculate \( f(9) \): \[ f(9) = 9^2 + 4(9) = 81 + 36 = 117 \] 3. The slope of the secant line is given by the formula: \[ \text{slope} = \frac{f(9) - f(3)}{9 - 3} = \frac{117 - 21}{9 - 3} = \frac{96}{6} = 16 \] So the slope of the secant line from \( x=3 \) to \( x=9 \) is \( 16 \). (B) Next, we need to find the equation of the secant line. We can use the point-slope form of the line equation. The slope we found is \( 16 \), and we can use the point \( (3, 21) \) to find the equation: \[ y - 21 = 16(x - 3) \] Distributing the slope: \[ y - 21 = 16x - 48 \] Adding \( 21 \) to both sides: \[ y = 16x - 27 \] Thus, the equation of the secant line from \( x=3 \) to \( x=9 \) is \( y = 16x - 27 \). (C) Now, let's find the slope of the secant line from \( x = 3 \) to \( x = 6 \). 1. Calculate \( f(6) \): \[ f(6) = 6^2 + 4(6) = 36 + 24 = 60 \] 2. The slope of the secant line is given by: \[ \text{slope} = \frac{f(6) - f(3)}{6 - 3} = \frac{60 - 21}{6 - 3} = \frac{39}{3} = 13 \] So the slope of the secant line from \( x=3 \) to \( x=6 \) is \( 13 \). (D) Now, let’s find the equation of the secant line from \( x=3 \) to \( x=6 \). Using the slope we've found (which is \( 13 \)) and the point \( (3, 21) \): \[ y - 21 = 13(x - 3) \] Distributing: \[ y - 21 = 13x - 39 \] Adding \( 21 \) to both sides: \[ y = 13x - 18 \] Thus, the equation of the secant line from \( x=3 \) to \( x=6 \) is \( y = 13x - 18 \). In summary: (A) Slope of secant line \( = 16 \) (B) Equation: \( y = 16x - 27 \) (C) Slope of secant line \( = 13 \) (D) Equation: \( y = 13x - 18 \)