Question

1. $f(x)=5x - 2$ (a) find the average rate of change from 1 to 3. (b) find an equation of the secant line containing $(1,f(1))$ and $(3,f(3))$.

Explanation:

Step1: Calculate f(1) and f(3)

First, find f(1):
\[

$$\begin{align*} f(1)&=5\times1 - 2\\ &=3 \end{align*}$$

\]
Then, find f(3):
\[

$$\begin{align*} f(3)&=5\times3 - 2\\ &=13 \end{align*}$$

\]

Step2: Calculate the average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here \(a = 1\), \(b=3\), \(f(1)=3\) and \(f(3)=13\).
\[

$$\begin{align*} \text{Average rate of change}&=\frac{f(3)-f(1)}{3 - 1}\\ &=\frac{13 - 3}{2}\\ &=5 \end{align*}$$

\]

Step3: Find the equation of the secant line

The slope - intercept form of a line is \(y=mx + c\), where \(m\) is the slope and \(c\) is the y - intercept. The slope \(m\) of the secant line passing through \((x_1,y_1)=(1,3)\) and \((x_2,y_2)=(3,13)\) is the average rate of change, so \(m = 5\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(1,3)\) and \(m = 5\):
\[

$$\begin{align*} y-3&=5(x - 1)\\ y-3&=5x-5\\ y&=5x-2 \end{align*}$$

\]

Answer:

(a) 5
(b) \(y = 5x-2\)