Question

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

Explanation:

Step1: Find \(f(3)\)

Substitute \(x = 3\) into \(f(x)=2x^{2}-5x - 8\).
\[

$$\begin{align*} f(3)&=2\times3^{2}-5\times3 - 8\\ &=2\times9-15 - 8\\ &=18-15 - 8\\ &=-5 \end{align*}$$

\]

Step2: Find \(f(5)\)

Substitute \(x = 5\) into \(f(x)=2x^{2}-5x - 8\).
\[

$$\begin{align*} f(5)&=2\times5^{2}-5\times5 - 8\\ &=2\times25-25 - 8\\ &=50-25 - 8\\ &=17 \end{align*}$$

\]

Step3: Calculate the slope of the secant line

The slope \(m\) of the secant line between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here \(x_1 = 3,y_1=f(3)=-5,x_2 = 5,y_2=f(5)=17\).
\[

$$\begin{align*} m&=\frac{f(5)-f(3)}{5 - 3}\\ &=\frac{17-(-5)}{2}\\ &=\frac{17 + 5}{2}\\ &=\frac{22}{2}\\ &=11 \end{align*}$$

\]

Answer:

11