Question

this table shows values that represent a quadratic function.
what is the average rate of change for this quadratic function for the interval from ( x = 4 ) to ( x = 6 )?
a. -20
b. 20
c. 10
d. -10

Explanation:

Step1: Recall the average rate of change formula

The average rate of change formula for a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 4\), \(b=6\).

Step2: Identify \(f(a)\) and \(f(b)\) from the table

When \(x = 4\), \(y=f(4)=-17\); when \(x = 6\), \(y = f(6)=-37\).

Step3: Substitute values into the formula

Substitute \(a = 4\), \(b = 6\), \(f(a)=-17\), \(f(b)=-37\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{-37-(-17)}{6 - 4}=\frac{-37 + 17}{2}\).

Step4: Simplify the expression

First, simplify the numerator: \(-37+17=-20\). Then, \(\frac{-20}{2}=-10\).

Answer:

D. -10