Question

what is the average rate of change for this quadratic function for the interval from $x = 1$ to $x = 3$?
text description for graph
a. 4
b. -4
c. 8
d. -8

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, \(a = 1\), \(b=3\).

Step2: Find the values of \(f(1)\) and \(f(3)\) from the graph

From the graph, when \(x = 1\), \(y=f(1)= - 3\) (by observing the \(y\) - coordinate of the point on the parabola at \(x = 1\)). When \(x = 3\), \(y=f(3)=-11\) (by observing the \(y\) - coordinate of the point on the parabola at \(x = 3\)).

Step3: Substitute the values into the formula

Substitute \(a = 1\), \(b = 3\), \(f(a)=-3\), \(f(b)=-11\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{-11-(-3)}{3 - 1}=\frac{-11 + 3}{2}=\frac{-8}{2}=-4\).

Answer:

B. -4