Question

for the function ( f(x)=2 x^{2}-5 x - 8 ), find the slope of the secant line between ( x = 3 ) and ( x = 5 ).

Explanation:

Step1: Find \( f(3) \) and \( f(5) \)

Substitute \( x = 3 \) into \( f(x)=2x^{2}-5x - 8 \):
\( f(3)=2\times3^{2}-5\times3 - 8=2\times9 - 15 - 8=18 - 15 - 8=-5 \)
Substitute \( x = 5 \) into \( f(x)=2x^{2}-5x - 8 \):
\( f(5)=2\times5^{2}-5\times5 - 8=2\times25 - 25 - 8=50 - 25 - 8 = 17 \)

Step2: Use the slope formula for the secant line

The slope formula for the secant line between two points \((x_1,f(x_1))\) and \((x_2,f(x_2))\) is \(m=\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here \(x_1 = 3\), \(x_2 = 5\), \(f(x_1)=-5\), \(f(x_2)=17\)
\(m=\frac{17-(-5)}{5 - 3}=\frac{17 + 5}{2}=\frac{22}{2}=11\)

Answer:

\(11\)