Question

on on the given intervals?

1. ( y = 5x - 3 ) on ( 1 leq x leq 7 )

the average rate of change of a linear function is constant. in this case, it is 5.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function \( y = f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\).

Step2: Identify \(a\), \(b\), \(f(a)\), \(f(b)\)

Here, \(a = 1\), \(b = 7\), \(f(x)=5x - 3\).
First, find \(f(1)\): \(f(1)=5(1)-3 = 2\).
Then, find \(f(7)\): \(f(7)=5(7)-3=35 - 3 = 32\).

Step3: Calculate average rate of change

Using the formula: \(\frac{f(7)-f(1)}{7 - 1}=\frac{32 - 2}{6}=\frac{30}{6}=5\).
Alternatively, for a linear function \(y = mx + c\), the average rate of change is the slope \(m\), which is \(5\) here.

Answer:

The average rate of change is \(5\).