Question

the function ( y = f(x) ) is graphed below. what is the average rate of change of the function ( f(x) ) on the interval ( -1leq xleq0 )?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-1\) and \(b = 0\).

Step2: Find \(f(-1)\) and \(f(0)\) from the graph

From the graph, when \(x=-1\), \(f(-1)=-4\) (by looking at the \(y\) - coordinate of the point on the graph with \(x=-1\)). When \(x = 0\), \(f(0)=-8\) (by looking at the \(y\) - coordinate of the point on the graph with \(x = 0\)).

Step3: Substitute into the formula

Substitute \(a=-1\), \(b = 0\), \(f(a)=-4\), and \(f(b)=-8\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{-8-(-4)}{0-(-1)}=\frac{-8 + 4}{0 + 1}\).

Step4: Simplify the expression

Simplify \(\frac{-8 + 4}{0 + 1}\). \(-8+4=-4\) and \(0 + 1=1\), so \(\frac{-4}{1}=-4\).

Answer:

\(-4\)