Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval ( 12 leq x leq 36 ).

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 12\) and \(b=36\). From the table, \(f(12)=34\) and \(f(36)=28\).

Step2: Substitute values into the formula

Substitute \(a = 12\), \(b = 36\), \(f(a)=34\), and \(f(b)=28\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{28 - 34}{36-12}\).

Step3: Simplify the expression

First, calculate the numerator: \(28-34=-6\). Then calculate the denominator: \(36 - 12 = 24\). So the fraction is \(\frac{-6}{24}\). Simplify \(\frac{-6\div6}{24\div6}\) (dividing numerator and denominator by their greatest - common divisor, which is 6).

Answer:

\(-\frac{1}{4}\)